Warm Up Find the unknown length for each right triangle with legs a and b and hypotenuse c. NO DECIMALS 5. b = 12, c =13 6. a = 3, b = 3 a = 5

Understand and use trigonometric relationships of acute angles in triangles. Determine side lengths of right triangles by using trigonometric functions. Objectives

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.

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.

Example 1: Finding Trigonometric Ratios Find the value of the sine, cosine, and tangent functions for θ. sin θ = cos θ = tan θ =

Find the value of the sine, cosine, and tangent functions for θ. sin θ = cos θ = tan θ =

Example 2: Finding Side Lengths of Special Right Triangles Use a trigonometric function to find the value of x. ° x = 37

Use a trigonometric function to find the value of x. °

Example 3: 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.

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 ≈ 41 The 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.

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.

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.

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 ≈ x Use a calculator to solve for x.

Step 3 Determine the overall height of the tree. x + 6 = = 150 The height of the tree is about 150 ft.

The reciprocals of the sine, cosine, and tangent ratios are also trigonometric ratios. They are trigonometric functions, cosecant, secant, and cotangent.

Example 5: Finding All Trigonometric Functions Find the values of the six trigonometric functions for θ. Step 1 Find the length of the hypotenuse θ a 2 + b 2 = c 2 c 2 = c 2 = 5476 c = 74 Pythagorean Theorem. Substitute 24 for a and 70 for b. Simplify. Solve for c. Eliminate the negative solution.

Example 5 Continued Step 2 Find the function values.

In each reciprocal pair of trigonometric functions, there is exactly one “co” Helpful Hint

Solve each equation. Check your answer. 1. Find the values of the six trigonometric functions for θ.

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

FRONT SIDE IS HW 5 BACK SIDE IS GRADED ASSIGNMENT DUE THURSDAY