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Holt Algebra 2
13-1 Right-Angle Trigonometry13-1 Right-Angle Trigonometry
Holt Algebra 2
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz
Holt Algebra 2
13-1 Right-Angle Trigonometry
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°
Holt Algebra 2
13-1 Right-Angle Trigonometry
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
Holt Algebra 2
13-1 Right-Angle Trigonometry
Understand and use trigonometric relationships of acute angles in triangles.
Determine side lengths of right triangles by using trigonometric functions.
Objectives
Holt Algebra 2
13-1 Right-Angle Trigonometry
trigonometric functionsinecosinetangentcosecantssecantcotangent
Vocabulary
Holt Algebra 2
13-1 Right-Angle Trigonometry
A trigonometric function is a function whose rule is given by a trigonometric ratio. A trigonometric ratio compares the lengths of two sides of a right triangle. The Greek letter theta θ is traditionally used to represent the measure of an acute angle in a right triangle. The values of trigonometric ratios depend upon θ.
Holt Algebra 2
13-1 Right-Angle Trigonometry
Holt Algebra 2
13-1 Right-Angle Trigonometry
The triangle shown at right is similar to the one in the table because their corresponding angles are congruent. No matter which triangle is used, the value of sin θ is the same. The values of the sine and other trigonometric functions depend only on angle θ and not on the size of the triangle.
Holt Algebra 2
13-1 Right-Angle Trigonometry
Example 1: Finding Trigonometric Ratios
Find the value of the sine, cosine, and tangent functions for θ.
sin θ =
cos θ =
tan θ =
Holt Algebra 2
13-1 Right-Angle Trigonometry
Check It Out! Example 1
Find the value of the sine, cosine, and tangent functions for θ.
sin θ =
cos θ =
tan θ =
Holt Algebra 2
13-1 Right-Angle Trigonometry
You will frequently need to determine the value of trigonometric ratios for 30°,60°, and 45° angles as you solve trigonometry problems. Recall from geometry that in a 30°-60°-90° triangle, the ration of the side lengths is 1: 3 :2, and that in a 45°-45°-90° triangle, the ratio of the side lengths is 1:1: 2.
Holt Algebra 2
13-1 Right-Angle Trigonometry
Holt Algebra 2
13-1 Right-Angle Trigonometry
Example 2: 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.
Holt Algebra 2
13-1 Right-Angle Trigonometry
Check It Out! Example 2
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.
Holt Algebra 2
13-1 Right-Angle Trigonometry
Example 3: Sports ApplicationIn 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.
Holt Algebra 2
13-1 Right-Angle Trigonometry
Make sure that your graphing calculator is set to interpret angle values as degrees. Press . Check that Degree and not Radian is highlighted in the third row.
Caution!
Holt Algebra 2
13-1 Right-Angle Trigonometry
Check It Out! Example 3 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.
Holt Algebra 2
13-1 Right-Angle Trigonometry
When an object is above or below another object, you can find distances indirectly by using the angle of elevation or the angle of depression between the objects.
Holt Algebra 2
13-1 Right-Angle Trigonometry
Example 4: 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.
Holt Algebra 2
13-1 Right-Angle Trigonometry
Example 4 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.
Holt Algebra 2
13-1 Right-Angle Trigonometry
Example 4 Continued
Step 3 Determine the overall height of the tree.
x + 6 = 144 + 6
= 150
The height of the tree is about 150 ft.
Holt Algebra 2
13-1 Right-Angle Trigonometry
Check It Out! Example 4
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°
Holt Algebra 2
13-1 Right-Angle Trigonometry
Check It Out! Example 4 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.
Holt Algebra 2
13-1 Right-Angle Trigonometry
Check It Out! Example 4 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.
Holt Algebra 2
13-1 Right-Angle Trigonometry
The reciprocals of the sine, cosine, and tangent ratios are also trigonometric ratios. They are trigonometric functions, cosecant, secant, and cotangent.
Holt Algebra 2
13-1 Right-Angle Trigonometry
Example 5: 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
c2 = 242 + 702
c2 = 5476
c = 74
Pythagorean Theorem.
Substitute 24 for a and 70 for b.
Simplify.
Solve for c. Eliminate the negative solution.
Holt Algebra 2
13-1 Right-Angle Trigonometry
Example 5 Continued
Step 2 Find the function values.
Holt Algebra 2
13-1 Right-Angle Trigonometry
In each reciprocal pair of trigonometric functions, there is exactly one “co”
Helpful Hint
Holt Algebra 2
13-1 Right-Angle Trigonometry
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
Holt Algebra 2
13-1 Right-Angle Trigonometry
Check It Out! Example 5 Continued
Step 2 Find the function values.
Holt Algebra 2
13-1 Right-Angle Trigonometry
Lesson Quiz: Part I
Solve each equation. Check your answer.
1. Find the values of the six trigonometric functions for θ.
Holt Algebra 2
13-1 Right-Angle Trigonometry
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