Math- Lesson 8-7: Unit 5 Review (Part -)
Trigonometric Functions sin cos A A SOH-CAH-TOA Some old horse caught another horse taking oats away. opposite ( length ) o sin A hypotenuse ( length ) h SOH adjacent ( length ) hypotenuse ( length ) cos A a h CAH tan A opposite ( length ) adjacent ( length ) tan A o a TOA
Trigonometric Functions (Mnemonics) SHA-CHO-CAO SHA CHO CAO
Sine Ratio What is the sine ratio of angle A? opp C sin A hyp 7 Sine ratio of angle A is 7 sin A 8 8 8 7 64 56 8 Sine ratio of angle B = B 7 4 sin 8 B 8 A 4
What do you notice about the trig ratios? B 5 4 sin A sin B 5 5 4 cos B cos A 5 5 C sin 4 A cos B A tan A 4 tan B 4
Solve the Triangle ma mb 90 4 mb 90 mb 47 mc 90 Must use trigonometry Solve for using either: (1) Pythagorean Theorem (more work) () Trigonometry (easy) Could you use mb sin A sin 4 to find? y=8. opp hyp B y C y 1 y 1sin 4 y 8. 47º cos 4 4º 1 =8.8 1 1cos 4 8.8 A
5 A 4º Solving a Right Triangle: If you don t know what trig. to use, write equations for all of them 47º C y sin 4º = y/5.4 B tan 4º = y/ y = 5 sin 4º =.4 cos 4º = /5 = 5 cos 4º =.7
A What if the angle is unknown? 1. Write the trig ratio. 5 sin 8. To undo the sine function, use the inverse of the sine function. 8 C 5 B sin 1 (sin ) sin sin 1 8.7 5 8 1 5 8
Angle of Elevation: angle above the horizon that the eye has to look up to see something. Angle of Depression: angle below the horizon that the eye has to look down at something.
The angle of elevation from the buoy to the top of the Barnegat Bay lighthouse 10 feet above the surface of the water is 5 º. Find the distance from the base of the lighthouse to the buoy. 1. Draw the picture. Write the equation.. Solve for the unknown variable. tan( 5 ) 10 10 5 º 10 1485. 9 ft tan(5 )
If the height of a building is 470 m and you are standing 100 m away from the building, find the angle of elevation to the top of the building. 470 100 470 tan( ) 100 º tan 1 470 100 78
base of a triangle: any side of a triangle. Height = Altitude base = side How many different ways are there to calculate the area of a triangle? A 1 * base* height three
The altitude of a triangle. A 1 * base* height Area formula: requires the use of matching altitudes and sides. Using segment AC as the base, requires the use of segment BD as the height.
Obtuse Triangle: has one angle that is obtuse (measures greater than 90 ) Two of the altitudes will be outside the triangle! You must etend a side to get a perpendicular intersection.
Areas of Obtuse Triangles: you must solve a right triangle to find the height. 1. Use linear pair theorem to find the angle opposite the height h.. Use SOHCAHTOA to obtain an equation with h in it. sin 7 h 7 7sin 7 h h 4.. Find area 7º 14º A 0.5*b*h A 0.5(5)(4.) A 10.5 units
Find the Area of the Triangle A 7.6 h 8 C 51º 10 48º B
There are two right triangles that can be used to solve for h. opp sin A hyp h sin(48) 8 h = 8*Sin(48º) h = 5.9 Area 0.5*b*h C Area = ½(10)(5.9) 7.6 51º Area = 9.5 square units A 10 h 8 48º B
Area of a Triangle B 5 Area 1 *base* height What is the base? What is the height? C 4 A For right triangles one of the legs is the base, the other leg is the height (or vice-versa). There is no point in finding the height from Angle C to side AB.
Vocabulary radian measure arc length radius Radian measure: the ratio of the arc length to the distance the arc is from the verte of the angle.
Vocabulary radian measure arc length radius Radian measure for a complete circle. radian measure of a circle circumference radius radian measure of a circle r r radian measure of a circle radians Units of radians = inches/inches Radian measure has no units! (nice)
Converting from Degrees to Radian Measure 140 140 60 o Write a proportion, solve for. 14 * 6 7 * 18 7 9 Converting from Radian Measure to degrees. 4 Write a proportion, solve for. *60 4* 4 60 o *90 4* *45 60 o o 15
Problem types you ll see: What is length of the subtended arc? r 5inches Write a proportion, solve for. arc angle circumference arc 10* * in arc 10* * in * arc 10 inches
What is length of the subtended arc? r 7 inches 50 Write a proportion, solve for. arc circumference arc * *7in 50 60 angle 60 5 arc ** *7*inches 6 arc 5 18 inches
Sector Area Problems What is the area of the sector? r = 8 ft area 0º Write a proportion, solve for. 64 * in 18 sector area circle area area *(8in) area 64 * in in 9 angle 60 0 60 1 18
Sector Area Problems Sector area =? Write a proportion, solve for. sector area angle circle area area *(6.5in) r = 6.5 ft area 1.7* in * area 1.7* in 44.* in
45-45-90 Right Triangle Is the triangle a) Obtuse b) Isosceles c) Scalene Solve for h using the Pythagorean theorem. a b c 1 1 c c c A 45º 45º 1 Label the side lengths of a similar 45-45-90 triangle using a scale factor of. A 45º 45º B C B C 1
Use scale factors or proportions to solve for the lengths of sides of similar 45-45-90 right triangles. Y 45 X 45º 1 45 6 Scale factor 45º 1
Solve for *(SF) 7 10 7 10 45 X 45 X Scale factor 7 5 7 10 7 5 45º 1 45º 1 Scale factor: a ratio of the lengths of two corresponding sides.
Building a 0-60-90 triangle. Start with an equilateral triangle whose side lengths are a convenient number ( ). What are the measures of the angles? 60º 60º 60º
Construct an angle bisector of the top angle. What lengths are the two segments formed at the bottom of the original triangle? Why? CPCTC (bottom legs are congruent so each is ½ the total bottom length. 0º 0º 60º 1 1 60º
We now have a 0-60-90 triangle. a b c a a 1 4 1 a 0º 60º 1
You have to remember this triangle. 1. The shortest side is opposite the smallest angle.. The longest side is opposite the largest angle. Which is a larger number; or? 1 1.7 0º 1 60º 1 4
We use similarity to solve for the missing sides. What are the measures c and a? 0º Scale factor 6 0º c 1 60º *( SF) ( SF) 6 6 a 60º ( SF) ( SF) a c