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Algebra 2
Lesson 1: Right Angle Trig.
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Warm Up
Given the measure of one of the acute angles in a right triangle, find the measure of the other acute angle.
1. 45° 2. 60° 3. 24° 4. 38°
45° 30°
66° 52°
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Warm Up Continued
Find the unknown length for each right triangle with legs a and b and hypotenuse c.
5. b = 12, c =13 6. a = 3, b = 3
a = 5
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Understand and use trigonometric relationships of acute angles in triangles.
Determine side lengths of right triangles by using trigonometric functions.
Objectives
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trigonometric functionsinecosinetangentcosecantssecantcotangent
Vocabulary
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SOH CAH TOA
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Example 1
Find the value of the sine, cosine, and tangent functions for θ.
sin θ =
cos θ =
tan θ =
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The reciprocals of the sine, cosine, and tangent ratios are also trigonometric ratios. They are trigonometric functions, cosecant, secant, and cotangent.
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Example 2: Finding All Trigonometric Functions
Find the values of the six trigonometric functions for θ.
Step 1 Find the length of the hypotenuse.
70
24
θ
a2 + b2 = c2
242 + 702 = c2
5476 = c2
74 = c
Pythagorean Theorem.
Substitute 24 for a and 70 for b.
Simplify.
Solve for c. Eliminate the negative solution.
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Example 2 Continued
Step 2 Find the function values.
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Example 3: Finding Side Lengths of Special Right Triangles
Use a trigonometric function to find the value of x.
°
x = 37
The sine function relates the opposite leg and the hypotenuse.
Multiply both sides by 74 to solve for x.
Substitute for sin 30°.
Substitute 30° for θ, x for opp, and 74 for hyp.
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Example 4
Use a trigonometric function to find the value of x.
The sine function relates the opposite leg and the hypotenuse.
Substitute 45 for θ, x for opp, and 20 for hyp.
°°
Substitute for sin 45°.
Multiply both sides by 20 to solve for x.
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Example 5 A skateboard ramp will have a height of 12 in., and the angle between the ramp and the ground will be 17°. To the nearest inch, what will be the length l of the ramp?
l ≈ 41The length of the ramp is about 41 in.
Substitute 17° for θ, l for hyp., and 12 for opp.
Multiply both sides by l and divide by sin 17°.
Use a calculator to simplify.
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Example 6: Sports Application
In a waterskiing competition, a jump ramp has the measurements shown. To the nearest foot, what is the height h above water that a skier leaves the ramp?
5 ≈ h
The height above the water is about 5 ft.
Substitute 15.1° for θ, h for opp., and 19 for hyp.
Multiply both sides by 19.
Use a calculator to simplify.
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Example 7: Geology Application
A biologist whose eye level is 6 ft above the ground measures the angle of elevation to the top of a tree to be 38.7°. If the biologist is standing 180 ft from the tree’s base, what is the height of the tree to the nearest foot?
Step 1 Draw and label a diagram to represent the information given in the problem.
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Example 7 Continued
Step 2 Let x represent the height of the tree compared with the biologist’s eye level. Determine the value of x.
Use the tangent function.
180(tan 38.7°) = x
Substitute 38.7 for θ, x for opp., and 180 for adj.
Multiply both sides by 180.
144 ≈ x Use a calculator to solve for x.
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Example 7 Continued
Step 3 Determine the overall height of the tree.
x + 6 = 144 + 6
= 150
The height of the tree is about 150 ft.
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Example 8
A surveyor whose eye level is 6 ft above the ground measures the angle of elevation to the top of the highest hill on a roller coaster to be 60.7°. If the surveyor is standing 120 ft from the hill’s base, what is the height of the hill to the nearest foot?
Step 1 Draw and label a diagram to represent the information given in the problem.
120 ft
60.7°
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Example 8 Continued
Use the tangent function.
120(tan 60.7°) = x
Substitute 60.7 for θ, x for opp., and 120 for adj.
Multiply both sides by 120.
Step 2 Let x represent the height of the hill compared with the surveyor’s eye level. Determine the value of x.
214 ≈ x Use a calculator to solve for x.
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Example 8 Continued
Step 3 Determine the overall height of the roller coaster hill.
x + 6 = 214 + 6
= 220The height of the hill is about 220 ft.
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In each reciprocal pair of trigonometric functions, there is exactly one “co”
Helpful Hint
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Find the values of the six trigonometric functions for θ.
Step 1 Find the length of the hypotenuse.
80
18
θ
a2 + b2 = c2
c2 = 182 + 802
c2 = 6724
c = 82
Pythagorean Theorem.
Substitute 18 for a and 80 for b.
Simplify.
Solve for c. Eliminate the negative solution.
Check It Out! Example 5
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Check It Out! Example 5 Continued
Step 2 Find the function values.
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Lesson Quiz: Part I
Solve each equation. Check your answer.
1. Find the values of the six trigonometric functions for θ.
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Lesson Quiz: Part II
2. Use a trigonometric function to find the value of x.
3. A helicopter’s altitude is 4500 ft, and a plane’s altitude is 12,000 ft. If the angle of depression from the plane to the helicopter is 27.6°, what is the distance between the two, to the nearest hundred feet?16,200 ft