Lesson 2: Right Triangle Trigonometry

Transcription

1 Lesson : Right Triangle Trigonometry lthough Trigonometry is used to solve many prolems, historially it was first applied to prolems that involve a right triangle. This an e extended to non-right triangles (hapter ), irles and irular motion, and a wide variety of appliations. s we shall see in the next unit, the six parts of any triangle ( sides and angles) are inherently linked through the proesses of Trigonometry. Given that the triangle is a right triangle we know: one angle is 90 ; the side the right angle is the longest in the triangle (and the smallest side is the smallest angle); the remaining two angles are omplementary, and so if we know one other angle we know all three angles in the triangle; the sides are related y Pythagoras Theorem, and so if we know any two sides, we an find the third side of the triangle. Using Trigonometry, if we are given a right triangle and the length of any two sides, we an determine the third side y using Pythagoras Theorem. This in turn is suffiient information to alulate the trigonometri ratios of the angles of the triangle and onsequently the measures of the angles..: THE RELTIONSHIP ETWEEN SIDES ND NGLES OF TRINGLE The sides of the triangle may also e named aording to their relationship to a given angle. In a right triangle, the is always the side the right angle. The side the angle is the alled the side; while the side whih forms one arm of the angle is alled the side. onsider the triangle Δ shown. In relation to the angle, o the side a is o the side is. DJENT HYPOTENUSE a OPPOSITE In relation to the angle, o the side a is OPPOSITE HYPOTENUSE o the side is. a DJENT NOTE: These relationships do not apply to the right angle, Right Triangle Trigonometry Page

2 onsider any two right triangles. If the triangles have one other equal angle, x say, then the third angle of eah triangle, (90 - x), must also e equal. Therefore the two triangles must e similar. y setion., sine the two triangles are similar we know that pairs of orresponding sides are in fixed ratio. PR QR QR PQ Taking these ratios one at a time: PR PQ (90 - x ) P (90 - x ) PR side to PQ x QR side to x PQ x R x Q PR side to x QR side to x Notie also these ratios for an e related to the angle x or to the omplement of the angle, (90 - x ): QR side to ( 90 x ) PQ and PR side to ( 90 x ) PQ To maintain onsisteny, note that PR QR gives us QR PR and so QR side to ( 90 x ) PR side to ( 90 x ) eause these fixed ratios are very important to us we assign them speial names, as you will see in the next setion..: THE DEFINITIONS OF THE TRIGONOMETRI RTIOS IN RIGHT TRINGLE sin a sin os os a tan a a tan a NOTE: Like all ratios, the trigonometri ratios do not have units. Whatever units are used to measure the sides are anelled out during the division proess. Right Triangle Trigonometry Page

3 Trigonometri Ratios of omplimentary ngles In a right triangle the two non-right angles must always sum to 90 and thus are always omplimentary. In the right triangle Δ, = 90, the angles and are omplimentary, and sin = os and os = sin The side is to and to ; the side a is to and to. Thus: side sin a side os and side os side sin Finding the Trigonometri Ratios in a Right Triangle Example : In Δ, = 90, a = and =. Find: a) the length of the, and ) the values of sin, os, tan. ) the values of sin, os, tan. First use Pythagoras Theorem to find the length of the : 9 For the, the side measures units and the side measures units. DJENT OPPOSITE HYPOTENUSE sin os tan For the, the side measures units and the side measures units. OPPOSITE DJENT HYPOTENUSE sin os tan NOTE: sin os and os sin Right Triangle Trigonometry Page

4 Example : In Δ, = 90, a = and =. Find (a) the length of the other leg (third side),and () the values of sin, os, tan. () the values of sin, os, tan. First, draw a diagram laeling the data given and the unknown values. Use Pythagoras Theorem to find the length of the third side of the triangle,. 0 0 NOTE : 0 For the, the side measures and the side measures. HYPOTENUSE OPPOSITE sin os tan DJENT rationalize the denominator: tan For the, the side measures and the side measures. HYPOTENUSE OPPOSITE DJ ENT sin os tan NOTE: sin os and os sin Right Triangle Trigonometry Page

5 Example : In ΔPQR, R = 90, p = and r =. Find the length of the other leg (third side),of the triangle and: (a) os Q () tan P Draw a diagram laeling the data given and the unknown values. Using Pythagoras Theorem to find q R q q q q P q Q For Q, the side measures unit; for P the side measures units and the side measures unit. R osq tan P P Q Example : In ΔXYZ, Z = 90, x = and y = 7. Find (a) the length of the z () tan Y () sin X z X Draw a diagram laeling the data given and the unknown values to e found. Using Pythagoras Theorem to find z: 7 z 9 7 z z z Y Z 7 Sustituting this value for z we an now find the trigonometri ratios: tany 7 sin X Y 7 X Z Right Triangle Trigonometry Page

6 Example : Using the alulator to approximate values In Δ, is a right angle, =. and =.. Find (a) the length of the third side of the triangle, orret to one deimal plae. () sin, os, and tan, approximated orret to the nearest thousandth.. a () sin, os, and tan, approximated orret to the nearest thousandth.. Using Pythagoras Theorem to find a: a a.. a.. a.. alulator entry: (. x. x ) = a a. 9 orret to one deimal plae Sustituting this value for in Δ we an find the six trigonometri ratios:..9 sin os tan sin os tan Right Triangle Trigonometry Page

7 Exerise. In eah right triangle Δ desried elow, = 90. Find the exat values of sin, os and tan.... In eah right triangle elow, find the exat value of sin, os, tan, and sin, os, and tan In Δ, = 90, a = 0, and = 0. Find the exat values of tan and os 8. In ΔPQR, R = 90, p =, and r =. Find the exat values of os P and tan Q 9. In ΔDEF, E = 90, d =.8, and e =.. Find the value of os D orret to three deimal plaes. 0. In ΔXYZ, X = 90, x =.8, and y = 7.. Find the value of tan Z orret to three deimal plaes. Right Triangle Trigonometry Page 7

8 .: EXT TRIGONOMETRI RTIOS FOR 0, ND 0 NGLES. Exat Trigonometri Ratios for 0 and 0. Sine the sides of any triangle are in the fixed ratio of 0 : : we may use this triangle to determine the exat trigonometri ratios for 0 and 0. From this triangle we see that: 0 sin0 os 0 tan 0 and sin 0 os 0 tan 0 Exat Trigonometri Ratios for The sides of any isoseles right triangle are in the fixed ratio of : :. From this triangle we see that sin os tan Exat Trigonometri Ratios for 0 and 90 0 With a little it of imagination we an oneptually derive the exat ratios for 0 and 90. onsider a right triangle, with another angle of 90. The third angle must e 0. If this triangle has one side of unit, the other side must e 0 units in length, and the will e unit. From this triangle we see that: sin 0 0 os 0 0 tan 0 0 and sin 90 0 os 90 0 tan 90 undefined 0 Summary: sin 0 os 0 tan 0 undefined Right Triangle Trigonometry Page 8

9 .: USING THE LULTOR TO FIND TRIGONOMETRI RTIOS Most alulators work in one of two ways when omputing the values for the trigonometri funtions. You need to e familiar with yours. First hek that the MODE of the alulator is degrees [DEG] and not radians [RD] LULTOR TYPE LULTOR TYPE (older style). Press SIN, OS or TN. Enter a numer in deimal degrees. Enter a numer in deimal degrees. Press SIN, OS or TN. Press equal/enter to get the value Note: the equals sign is not needed here. Round the value to the required numer of deimal plaes. Example : Find the value of, approximated orret to the nearest thousandth: (a) sin 7 () os () tan.9 (a) Find sin 7 to the nearest thousandth Enter: alulator Type alulator Type sin 7 = 7 sin Display: the numer of deimal plaes given in the alulator s answer will depend on your alulator:.998 Rounding: we want the nearest thousandth we approximate orret to three deimal plaes:.9 98 and sine the digit in the fourth deimal plae is 9, we round the in the third plae up to. nswer: sin () Find os to the nearest thousandth Enter: alulator Type alulator Type os = os Display:.78 Right Triangle Trigonometry Page 9

10 Rounding: approximate orret to three deimal plaes.7 8 sine the digit in the fourth deimal plae is, leave the numer in the third plae as. nswer: os 0.7 () Find tan.9 to the nearest thousandth Enter: alulator Type alulator Type tan.9 =.9 tan Display: Rounding: approximate orret to three deimal plaes sine the digit in the fourth deimal plae is, round the in the third plae up to. nswer: tan. Example : Draw a diagram of a right triangle whih has an angle of. Find a) the size of the third angle, and ) the sine, osine and tangent ratios of eah angle orret to the nearest thousandth. 7 a) Sine the non-right angles of a right triangle are omplementary (i.e. add to 90 ) the third angle is: 90 - = 7 (Or, sine the angle sum of a triangle is 80, the third angle is: = 7 ) ) Using a alulator: 0 sin os tan 0. 0 sin os tan 7. Right Triangle Trigonometry Page 0

11 Exerise. Questions 0: Use a alulator to find eah of the following. pproximate all answers orret to the nearest thousandth (three deimal plaes.) ) sin ) tan ) tan 87. ) os 7 ) os 8.8 ) sin 8 7) os.9 8) sin. 9) tan ) tan 8 Questions - : Use a alulator to find the value of eah of following. pproximate all answers orret to the nearest thousandth (three deimal plaes.) NOTE: In eah ase, the trigonometri funtions are ofuntions of one another, and the angles are omplementary angles. ) sin, os 77 ) sin, os 8 ) sin 0, os 90 ) sin 90, os 0 Questions - 7: Draw a diagram of a right triangle whih has the given angle. Find a) the size of the third angle, and ) the sine, osine and tangent ratios of eah angle orret to the nearest thousandth. ) 0 ).7 7) Questions 8 0: omplete the following tale using the values omputed and then use these values to answer the questions elow y ompleting the sentene sin os tan undefined 8) s the aute angle gets larger, the sine of the angle gets and the osine of the angle gets 9) The value of the sine and osine of an angle always lies etween the numers and 0) The smallest value of the tangent of an angle is and the iggest value is [HINT: In the last question, use your alulator to look at the values of tan 89, tan 89.9, tan 89.99, tan , tan , et] Right Triangle Trigonometry Page

12 .: USING TRIGONOMETRI RTIOS TO FIND N NGLE Most alulators work in one of two ways when omputing the angle given the value of the trigonometri ratio. You need to e familiar with yours. gain, first hek that the MODE of the alulator is degrees [DEG] and not radians [RD] NOTTION: If If If sin x os x tan x then then then sin x os x tan x LULTOR TYPE LULTOR TYPE (older style). Press INV or ND utton. Enter the value of the ratio. Press SIN, OS or TN. Press INV or ND utton. Enter the value of the ratio. Press SIN, OS or TN. Press equal/enter to get the value Note: the equals sign is not needed. Round the resulting angle to the required numer of deimal plaes. Example : Find the angle, orret to the nearest tenth of a degree, given the ratio. (a) sin = 0. () os = 0. () tan θ=. (a) If sin = 0., find the angle orret to the nearest tenth of a degree, given the ratio. Enter: alulator Type alulator Type Use the INV or ND utton Display: INV sin 0. = 0. INV sin the numer of deimal plaes given in the alulator s answer will depend on your alulator: 7.99 Rounding: we want the nearest tenth of a degree we approximate orret to one deimal plae: and sine the digit in the seond deimal plae is, whih is less than, we leave the numer in the first deimal plae as. nswer: If sin = 0. then = 7. hek: Using the alulator, sin 7. = Right Triangle Trigonometry Page

13 () If os = 0., find the angle orret to the nearest tenth of a degree, given the ratio. Enter: alulator Type alulator Type Use the INV or ND utton Display: INV os 0. = 0. INV os the numer of deimal plaes given in the alulator s answer will depend on your alulator: Rounding: we want the nearest tenth of a degree we approximate orret to one deimal plae: Sine the digit in the seond deimal plae is 9, whih is more than, we round the 9 in the first deimal plae to 0, whih will round 8.9 to 8.0. nswer: If os = 0. then = 8.0 NOTE: To orretly answer this question, we should write 8.0 not 8 in order to indiate that this angle is orret to the nearest tenth of a degree. hek: Using the alulator, os 8 = () If tan θ=., find the angle θorret to the nearest tenth of a degree, given the ratio. NOTE: θis the greek letter theta, ommonly used in trigonometry to represent an angle. Enter: alulator Type alulator Type ND Use the INV or utton Display: INV tan. =. INV tan Rounding: Round the 8 in the first deimal plae to 9 nswer: If tan θ=. then θ= 7.9 hek: Using the alulator, tan 7.9 =.000. This is not. as we would like. However, onsidering the alternatives, tan 7.0 =.708.7, tan 7.9 =.0. and tan 7.8 =.07.0, so we have the losest approximation to one tenth of a degree. Right Triangle Trigonometry Page

14 Exerise. Questions 8: Use a alulator to find eah of the following. pproximate all answers as instruted. ), to the nearest degree, if sin ), to the nearest tenth of a degree if, tan. ) P, to the nearest degree if osp 0. 0 ) if os ), to the nearest degree, if os0. 89 ), if sin 7), to the nearest tenth of a degree, if tan ) if tan Right Triangle Trigonometry Page

15 .: SOLVING RIGHT TRINGLES The three sides and three angles of any triangle are related through the trigonometri ratios, and we are ale to use trigonometry to determine the measure of the remaining omponents if we are given any three fats aout the triangle. In this setion we are solving right triangles. That is, we know that the triangle has one angle that of 90. Given this and: o the length of two sides, or o the size of one angle and the length of one side, we an find the remaining sides and angles of the triangle. In hapter we will extend our theory to non-right triangles. Solve the Right Triangle Given One ngle and One Side Example : Find the remaining sides and angles of the triangle Δ, given that is a right angle, = 0 and = 0 m. The first step is always to draw a diagram of the triangle. Lael the diagram with the information given and the unknowns to e found. In this example we know, and. We need to find, a, and a 0 0 To find we an use the fat that the two aute angles of the triangle are omplementary, To find a and we need to identify their relative positions in a trigonometri ratio of whih we know two out of the three omponents. To find a, we an use either os (sine a is the side to the given angle, and we 0 know the ) or sin (sine a is the side to the alulated angle, and we know the ). a 0 Using os : os Sustituting the known values for and, a os0 0 Multiplying oth sides of the equation y 0: Using a alulator a 0os 0 a 7.0 alulator: 0 x os 0 = Sine the least numer of signifiant figures in the data is three, answers in this prolem should e rounded orret to three signifiant figures: a 7. 7 Right Triangle Trigonometry Page

16 LTERNTE: Find a using sin : sin Sustituting the known values of and : a sin0 0 Multiplying oth sides y0 : a 0 sin0 Using a alulator : a 7. 7 alulator: 0 x sin 0 = To find, we an use either sin (sine is the side to the given angle, and we know the ) or os (sine is the side to the given angle, and we know the ), or we ould use Pythagoras Theorem. Using sin : sin Sustituting the known values of and : sin 0 0 Multiplying oth sidesy 0: 0sin 0 Usinga alulator : a. LTERNTE: Find using os : os hptotenuse Sustituting the known values of and : os0 0 Multiplying oth sidesy 0: alulator: 0os 0 Usinga alulator : 0 x os 0 = a. alulator: 0 x sin 0 = LTERNTE: We ould also find using Pythagoras Theorem. This is the least appealing method eause it uses an approximated alulation rather than an exat measure. NOTE: While heking our alulations, notie that the smallest side, =. m, is the smallest angle, = 0 ; the middle side, a = 7.7 m, is the middle angle, = 0, and the, = 0 m, is the longest side of the triangle. Right Triangle Trigonometry Page

17 Example : Solve the triangle Δ, given is a right angle, = 7. and a =. units. To find : To find tan is hosen instead of tan only eause it results in an slightly easier manipulation: tan tan... tan.. To find : sin a sin. sin NOTE: Sine the least numer of signifiant figures in the data is three, answers in this prolem should e rounded orret to three signifiant figures. alulator:. sin 7. = Example : Find the exat values of the remaining sides and angles of Δ, given is a right angle, = and a = units. Sine one angle of Δ is we know that the triangle is an isoseles triangle with two angles of ; the sides these angles are equal, and oth units Therefore = and =. In fat we also know that the sides are in the fixed ratio of :: and do not need to use trigonometri ratios to find in this ase. However to demonstrate the use of trigonometry, we will use the sine ratio to find. To find : sin sin sin We know sin, hene Right Triangle Trigonometry Page 7

18 Solve the Right Triangle Given the Length of Two Sides Example : Solve the triangle Δ, given is a right angle, a = m and = m. Draw a diagram and lael all known and unknown values. To find sin sin sin 7 The easiest method to find is to use the fat that it is the omplement of lternately, we an use the osine ratio: os alulator: INV sin ( ) = os os alulator: INV os ( ) = The easiest method to find is to use Pythgoras Theorem or etter yet, to reognize the missing element of the {,, } Pythagorean Triad to find that the exat measure of is units. lternately, we an use the sine, osine or tangent ratios to find : tan tan7 Re arranging : tan 7 This is the least appealing method eause we are using an approximated alulation when the more aurate given data ould e used. Right Triangle Trigonometry Page 8

19 Example : Solve the triangle Δ, given is a right angle, a =. m and =. m. Sine we are given the and sides for, we an use the tan ratio to find the angle. tan. tan.. tan.. 70 Sine and are omplementary: Using Pythagoras Theorem, we an find We an use the tangent ratio to find : tan. tan.. tan Solve the Right Triangle Given a Trigonometri Ratio Example : Find the sides and angles of a triangle ΔPQR, where P is a right angle and 9 sin R. 0 9 Sine sin R and sin R 0 9 sin R 0 then we an use That is, sine we know that the ratio of the sides is 9 to 0, we an set the side R as 9 units and the of the triangle as 0 units. Draw a diagram and lael verties and sides. y Pythagoras Theorem, 9 q 0 Sine 9 sin R 0 q q 9 then, 9 R sin 0 R Q and R are omplementary so: Q 90 Q 77 Q 9 P q 0 R Right Triangle Trigonometry Page 9

20 Example 7: In right triangle Δ, where is a right angle, find os given that tan. Sine tan and tan then for in Δ t this point, a diagram is very useful! We need to find os. os Using Pythagoras Theorem to find : Therefore os Right Triangle Trigonometry Page 0

21 ppliations Example 9: Find the altitude of an isoseles triangle whih has a ase of length 8 m and ase angles measuring. We need to find h. Sine h is the side the angle and we have the measure of the side to we will use the tangent ratio. h h tan h tan Sine we need to approximate orret to signifiant figures, the height of the triangle is alulated as 0 m. Example 0: ft ladder leans against a wall and reahes to a height of ft. What angle does the ladder form with the wall? Draw a diagram. Let e the angle os os ft ft Therefore the ladder makes a angle with the wall. Example : satellite is oriting 7 miles aove the earth s surfae. (See diagram) When it is diretly aove the point T, the angle S is found to e 7.. Find the radius of the earth. Let the radius of the earth e r miles. Then: r r sin 7. r 7 7 sin 7. r r sin7. 7sin 7. r 7sin 7. r r sin7. r 7 sin7. r sin7. sin7. 7. Radius, r S 7 mi T (r + ) mi r r 00.8 Hene the radius of the earth is approximately 000 miles (orret to three signifiant figures.) Right Triangle Trigonometry Page

22 ngles of Elevation and Depression ngles of Elevation and Depression are measured relative to the oserver. n imaginary line drawn from the eye of the oserver to the ojet eing oserved is alled the line of sight. The horizontal is the line of sight to an ojet diretly in front of, neither aove nor elow, the eye of the oserver. If the ojet eing oserved is aove the horizontal, then the angle etween the line of sight and the horizontal is alled the angle of elevation. (nother way to think of this is the angle the oserver would need to look up to the ojet.) If the ojet is elow the horizontal, then the angle etween the line of sight and the horizontal is alled the angle of depression. (nother way to think of this is the angle the oserver would need to look down to the ojet.) HORIZONTL ngle of Elevation ngle of Depression Note that the angle of elevation or depression is always measured from the oserver to the ojet. HORIZONTL Example : To determine the height of a tree, a student oserves that it asts a ft shadow when the angle of elevation of the sun (from the top of the shadow) is. Find the height of the tree. We need to find h, the height of the tree. First draw a diagram. h h The top of the tree, ase of the tree/foot of the shadow, and tip of the shadow form the three verties of a right triangle. We know one angle and two sides of the triangle: the angle of elevation, the length of the shadow whih is an side of the angle; and want to find the height of the tree, whih is the side of the triangle to the angle. Thus we use the tangent ratio. h tan h tan h 78 The tree is approximately 78 ft in height. Right Triangle Trigonometry Page

23 Example : n airplane is flying at an altitude of ft, diretly aove a straight streth of highway along whih a ar and a us are traveling towards eah other. The vehiles are on sides of the airplane, the ar at an angle of depression of.7 and the us at an angle of depression of. from the plane. How far apart are the vehiles to the nearest tenth of a mile? Let the ar e x ft and the us e y ft from the point whih the airplane is flying diretly aove. Then the ar and the us are (x + y) ft apart. Sine we have the measure of the angle and its side, and we wish to find the side, we use the tangent ratio. x tan. 7 x tan. 7 x.7. ft y y tan. y tan. x y tan. 7 tan. 0 ft 0 miles miles tan. 7 tan. Hene the vehiles are approximately. miles apart. Example : From a point.0 meters from the ase of a uilding, the angle of elevation to the top of the uilding is.. The angle of elevation from the same point to the tip of a flagpole on top of this uilding is 8.. What is the height of the flagpole? Let the uilding e h meters and the flagpole e x meters in height. Then: and h tan. h tan. x h tan7. x h tan8. x tan8. h Sustituting for h: x tan8. h tan8. tan. tan8. tan Hene the pole is approximately 9. m tall Right Triangle Trigonometry Page. 7. m x h

24 Exerise.. Solve the triangle Δ, given that is a right angle, and: (a) a = 0 and = () = 8 and = () a =. and =. (d) = and = 7 (e) = 8. and = 8.8 (f) = and =. Find the value of the side laeled x orret to the nearest tenth. (a) () () 7 x x.. x (d) (e) x 7 (f) 8 x 7. x. Find the angle orret to the nearest degree. (a) () 8 () 8.. (d) (e) (f) 7... Solve the right triangle Q (a) () () 0 R Y X Z P. Solve for the exat values of the right triangle P (a) () R () 0 8 M 0 N S T Right Triangle Trigonometry Page

25 . ladder 0 ft in length reahes 9 ft up a wall against whih it leans. Find the angle, to the nearest degree, that the ladder makes with the wall. 7. road up a hill makes an angle of. with the horizontal. If this road is. miles long, how high is the hill, to the nearest hundred feet? 8. When the angle of elevation of the sun is 7 a uilding asts a shadow of ft. How tall is the uilding the nearest foot? 9. ft man asts a shadow that is ft long. What is the angle of elevation of the sun, to the nearest degree? 0. The irle shown has a radius of r, and a enter at. If the distane DE = m, find the radius of the irle to the nearest entimeter. r r r D E E F. The ue shown has a side length of 0 m. Find the angle formed y the diagonals G and DG (orret to the nearest tenth of a degree.) 0 H 0 G D 0. kite string is extended ft in length when the kite makes an angle of elevation of. with the ground. Find the altitude of the kite to the nearest foot. h ft.. To measure the height of a tower aross a freeway, a student takes two measurements. She stands diretly aross from the point at the foot of the tower, and finds that the angle of elevation to the top of the tower is.. She then walks 0 ft parallel to the freeway (at a right angle to the point at whih she took the measure) and then finds that the angle from her new loation to the ase of the tower is 87.. Using this information, find the height of the tower orret to the nearest foot. h ft Right Triangle Trigonometry Page

26 . hot air alloon is floating aove a straight streth of highway. To estimate how high aove the ground the alloon is floating, the passengers of the alloon take measurements of a ar elow them. They assume that the ar is traveling at 0 miles per hour. One minute after the ar passes diretly elow the alloon they take a earing on the ar and find that the angle of depression to the ar is. Find the altitude of the alloon to the nearest 00 ft. h. man is standing 0 ft from a painting. He noties that the angle of elevation from his eyes to the top of the painting is 8 and the angle of depression to the ottom of the painting is. Find the height of the painting to the nearest tenth of a foot. 8 0 ft x. person standing on hill sees a tower that she knows to e 0 ft high. She oserves that the angle of elevation to the top of the tower is, while the angle of depression to the foot of the tower is. How far is she from the tower, orret to the nearest foot? x ft 0 ft 7. To estimate the height of a partiular mountain, the angle of elevation to the top of the mountain is measured to e ft loser to the mountain the angle of elevation is found to e. What is the height of the mountain to the nearest hundred feet? h 8 00 ft 8. Find the dimensions of the sheet of paper needed to draw an otagon of side m, to the nearest entimeter. Right Triangle Trigonometry Page