Solving Right Triangles How do you solve right triangles?
The Trigonometric Functions we will be looking at SINE COSINE TANGENT
The Trigonometric Functions SINE COSINE TANGENT
SINE Pronounced sign
TANGENT Pronounced tan-gent
COSINE Pronounced co-sign
Greek Letter q Pronounced theta Represents an unknown angle
What is Trigonometry? Trigonometry is the study of how the sides and angles of a triangle are related to each other. It's all about triangles!
Right Triangle Opposite Hypotenuse Adjacent q A
Same Right Triangle Different Angle B q Adjacent Hypotenuse Opposite
Trig Definitions: Sine = opposite/hypotenuse Cosine = adjacent/hypotenuse Tangent = opposite/adjacent Cosecant = hypotenuse/opposite Secant = hypotenuse/adjacent Cotangent = adjacent/opposite
x,y r y O x 2 2 2 r = x + y
Definitions of Trig Functions Sin O = y / r Cos O = x / r Tan O = y / x Csc O = r / y Sec O = r / x Cot O = x / y
The Unit Circle Radius = 1
30, 60, 90 Is a special kind of triangle. y x,y 1 1/2 30 3/2 x 2 2 2 r = x + y
45, 45, 90 Is a special kind of triangle. y x,y 1 /2/2 45 2/2 x 2 2 2 r = x + y
Finding sin, cos, and tan. Just writing a ratio.
Find the sine, the cosine, and the tangent of theta. Give a fraction. 35 37 sinq opp hyp 35 37 12 q cos q adj hyp 12 37 Shrink yourself down and stand where the angle is. tanq opp adj 35 12 Now, figure out your ratios.
Find the sine, the cosine, and the tangent of theta 8.2 24.5 q 23.1 sinq opp hyp 8.2 24.5 Shrink yourself down and stand where the angle is. cos q adj hyp 23.1 24.5 Now, figure out your ratios. tanq opp adj 8.2 23.1
Solving Right Triangles For Angles ( Theta )
If you know the sine, cosine, or tangent of an acute angle measure, you can use the inverse trigonometric functions to find the measure of the angle.
Calculating Angle Measures from Example 4 Trigonometric Ratios Use your calculator to find each angle measure to the nearest tenth of a degree. A. cos -1 (0.87) B. sin -1 (0.85) C. tan -1 (0.71) cos -1 (0.87) 29.5 sin -1 (0.85) 58.2 tan -1 (0.71) 35.4
Inverse trig functions: Ex: Use a calculator to approximate the measure of the acute angle. Round to the nearest tenth. 1. tan A = 0.5 2. sin A = 0.35 3. cos A = 0.64 m A 1 tan (0.5) m A 1 sin (0.35) m A 1 cos (0.64) 26.6 20.5 50.2
USING TRIG RATIOS TO FIND A MISSING SIDE
To find a missing SIDE 1. Draw stick-man at the given angle. 2. Identify the GIVEN sides (Opposite, Adjacent, or Hypotenuse). 3. Figure out which trig ratio to use. 4. Set up the EQUATION. 5. Solve for the variable.
1. H Problems match the WS. Where does x reside? If you see it up high then we MULTIP LY! A x cos15 9 9 cos15 x 8.7 x
2. Problems match the WS. H O Where does x reside? If X is down below, The X and the angle will switch sin50 9 x SLIDE & DIVIDE x 9 sin50 x 11.7
3. H Problems match the WS. cos51 10 10 x A x x 15.9 cos 51
Solving Right Triangles For All Six Parts
Every right triangle has one right angle, two acute angles, one hypotenuse, and two legs. To SOLVE A RIGHT TRIANGLE means to find all 6 parts. To solve a right triangle you need.. 1 side length and 1 acute angle measure -or- 2 side lengths
Given one acute angle and one side: To find the missing acute angle, use the Triangle Sum Theorem. The Triangle Sum Theorem states that the three interior angles of any triangle add up to 180 degrees. To find one missing side length, write an equation using a trig function. To find the other side, use another trig function or the Pythagorean Theorem: a 2 + b 2 = c 2. Note: c is the longest side of the triangle a and b are the other two sides
Solve the right triangle. Round decimal answers to the nearest tenth. GUIDED PRACTICE Example 1 Find m B by using the Triangle Sum Theorem. 180 o = 90 o + 42 o + m B 48 o = m B A 70 42 o 48 o Approximate BC by using a tangent ratio. C B tan 42 o = BC Approximate AB by using a cosine ratio. 70 70 cos 42 o = 70 tan 42 o AB ANSWER = BC AB cos 42 o = 70 70 0.9004 BC 70 The angle measures are 63.0 BC AB = cos 42 o 42 o, 48 o, and 90 o. The 70 side lengths are 70 feet, AB 0.7431 about 63.0 feet, and AB 94.2 about 94.2 feet.
Solve a right triangle that has a 40 o angle and a 20 GUIDED PRACTICE inch hypotenuse. Example 2 Find m X by using the Triangle Sum Theorem. 180 o = 90 o + 40 o + m X 50 o = m X Approximate YZ by using a sine ratio. sin 40 o = XY 20 20 sin 40 o = XY 20 0.6428 XY 12.9 XY Approximate YZ by using a cosine ratio. YZ cos 40 o = 20 20 cos 40 o = YZ 20 0.7660 YZ 15.3 YZ Y X 50 o ANSWER 20 in 40 o The angle measures are 40 o, 50 o, and 90 o. The side lengths are 12.9 in., about 15.3 in., and 20 in. Z
Solve the right triangle. Round to the nearest tenth. Example 3 p q cos53 sin53 30 30 p 18.1 q 24.0 P + R + Q = 180 P = 180-90-53 P = 37 m Q 53 m R 90 m P 37 PQ 30 PR 24.0 QR 18.1
Solve the right triangle. Round decimals to the nearest tenth. Example 5 Example 4 m P 90 37 53 m T 90 24 66 PQ sin37 22 PQ 13.2 QR cos37 22 QR 17.6 TR tan24 33 TR 14.7 33 cos24 AT AT 36.1
Solving Right Triangles Example 6 Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree. Method 1: By the Pythagorean Theorem, Method 2: RT 2 = RS 2 + ST 2 (5.7) 2 = 5 2 + ST 2 Since the acute angles of a right triangle are complementary, m T 90 29 61. Since the acute angles of a right triangle are complementary, m T 90 29 61., so ST = 5.7 sinr.
Example 7 Solve the right triangle. Round decimals the nearest tenth. Use Pythagorean Theorem to find c c c 2 3 2 2 3 3.6 Use an inverse trig function to find a missing acute angle m A tan ( ) 56.3 2 Use Triangle Sum Theorem to find the other acute angle 1 3 m B 90 56.3 33.7 AB 3.6 BC 3 AC 2 m A 56.3 m B 33.7 m C 90
Example 8 PN 11 18 2 2 2 PN 21.9 Pythagorean Theorem A2 + b2 = c2 Where c is the hypotenuse. 1 11 m N tan ( ) 31.4 18 m P 90 31.4 58.6
Example 9 23 TU 7 2 2 2 TU 21.9 m S cos ( ) 72.3 23 1 7 m U 90 72.3 17.7
Trig Application Problems MM2G2c: Solve application problems using the trigonometric ratios.
Depression and Elevation angle of depression horizontal angle of elevation horizontal
9. Classify each angle as angle of elevation or angle of depression. Angle of Depression Angle of Elevation Angle of Elevation Angle of Depression
Example 10 Over 2 miles (horizontal), a road rises 300 feet (vertical). What is the angle of elevation to the nearest degree? 5280 feet 1 mile tanq 300 10,560 q 2
Example 11 The angle of depression from the top of a tower to a boulder on the ground is 38º. If the tower is 25m high, how far from the base of the tower is the boulder? Round to the nearest whole number. tan38 25 x x 32meters
Example 12 Find the angle of elevation to the top of a tree for an observer who is 31.4 meters from the tree if the observer s eye is 1.8 meters above the ground and the tree is 23.2 meters tall. Round to the nearest degree. tanq 21.4 31.4 q 34
Example 13 A 75 foot building casts an 82 foot shadow. What is the angle that the sun hits the building? Round to the nearest degree. tanq 82 75 q 48
Example 14 A boat is sailing and spots a shipwreck 650 feet below the water. A diver jumps from the boat and swims 935 feet to reach the wreck. What is the angle of depression from the boat to the shipwreck, to the nearest degree? 650 si nq 935 q 44
Example 15 A 5ft tall bird watcher is standing 50 feet from the base of a large tree. The person measures the angle of elevation to a bird on top of the tree as 71.5. How tall is the tree? Round to the tenth. tan71.5 x 50 x 154.4feet
Example 16 A block slides down a 45 slope for a total of 2.8 meters. What is the change in the height of the block? Round to the nearest tenth. si n45 x 2.8 q 2meters
Example 17 A projectile has an initial horizontal velocity of 5 meters/second and an initial vertical velocity of 3 meters/second upward. At what angle was the projectile fired, to the nearest degree? tanq 3 5 q 31
Example 18 A construction worker leans his ladder against a building making a 60 o angle with the ground. If his ladder is 20 feet long, how far away is the base of the ladder from the building? Round to the nearest tenth. x cos60 x 10feet 60