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Lesson Menu Five-Minute Check (over Chapter 3) Then/Now New Vocabulary Key Concept: Trigonometric Functions Example 1:Find Values of Trigonometric Ratios Example 2:Use One Trigonometric Value to Find Others Key Concept:Trigonometric Values of Special Angles Example 3:Find a Missing Side Length Example 4:Real-World Example: Find a Missing Side Length Key Concept:Inverse Trigonometric Functions Example 5:Find a Missing Angle Measure Example 6:Real-World Example: Use an Angle of Elevation Example 7:Real-World Example: Use Two Angles of Elevation or Depression Example 8:Solve a Right Triangle

Over Chapter 3 5–Minute Check 1 A. Sketch the graph of f (x) = –2 x + 3. A. B. C. D.

Over Chapter 3 5–Minute Check 1 B. Analyze the graph of f (x) = –2 x + 3. Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing.

Over Chapter 3 5–Minute Check 1 A. B. C. D.

Over Chapter 3 5–Minute Check 2 A. Consider the table shown at the right. Make a scatter plot. A. B. C. D.

Over Chapter 3 5–Minute Check 2 B. Consider the table shown at the right. Find an exponential function to model the data. A.y = 2.05(3.28) x B.y = 43.47x 3 – x x – C.y = x 2 – x D.y = 0.99(x) 3.28

Over Chapter 3 5–Minute Check 2 C. Consider the table shown at the right. Find the value of the model at x = 20. A.4.26 x B.254,354 C.45,966 D.18,324

Over Chapter 3 5–Minute Check 3 Solve log 3 5x – log 3 (x + 3) = log 3 4. A.–3 B. C.3 D.12

Then/Now You evaluated functions. (Lesson 1-1) Find values of trigonometric functions for acute angles of right triangles. Solve right triangles.

Vocabulary trigonometric ratios trigonometric functions sine cosine tangent cosecant secant cotangent reciprocal function inverse trigonometric function inverse sine inverse cosine inverse tangent angle of elevation angle of depression solve a right triangle

Key Concept 1

Example 1 Find Values of Trigonometric Ratios Find the exact values of the six trigonometric functions of θ. The length of the side opposite θ is 33, the length of the side adjacent to θ is 56, and the length of the hypotenuse is 65.

Example 1 Find Values of Trigonometric Ratios Answer:

Example 1 Find the exact values of the six trigonometric functions of θ. A. B. C. D.

Example 2 Use One Trigonometric Value to Find Others Begin by drawing a right triangle and labeling one acute angle . Because sin  =, label the opposite side 1 and the hypotenuse 3. If, find the exact values of the five remaining trigonometric functions for the acute angle .

Example 2 Use One Trigonometric Value to Find Others By the Pythagorean Theorem, the length of the leg adjacent to 

Example 2 Use One Trigonometric Value to Find Others Answer:

Example 2 If tan  =, find the exact values of the five remaining trigonometric functions for the acute angle . A. B. C. D.

Key Concept 3

Example 3 Find a Missing Side Length Find the value of x. Round to the nearest tenth, if necessary.

Example 3 Find a Missing Side Length Because you are given an acute angle measure and the length of the hypotenuse of the triangle, use the cosine function to find the length of the side adjacent to the given angle. Cosine function Use a calculator ≈ x Multiply each side by 7. θ = 35°, adj = x, and hyp = 7

Example 3 Find a Missing Side Length Answer: about 5.7 Therefore, x is about 5.7. Simplify. x = 5.73 CheckYou can check your answer by substituting x = 5.73 into.

Example 3 Find the value of x. Round to the nearest tenth, if necessary. A.4.6 B.8.1 C.9.3 D.10.7

Example 4 Find a Missing Side Length SPORTS A competitor in a hiking competition must climb up the inclined course as shown to reach the finish line. Determine the distance in feet that the competitor must hike to reach the finish line. (Hint: 1 mile = 5280 feet.)

Example 4 Answer:about 5864 ft Find a Missing Side Length Tangent function So, the competitor must hike about 5864 feet to reach the finish line. Use a calculator. Multiply each side by θ = 48°, opp = x, and adj = 5280 An acute angle measure and the adjacent side length are given, so the tangent function can be used to find the opposite side length.

Example 4 WALKING Ernie is walking along the course x, as shown. Find the distance he must walk. A ft B ft C.69.5 ft D.8.5 ft

Key Concept 5

Example 5 Find a Missing Angle Measure Use a trigonometric function to find the measure of θ. Round to the nearest degree, if necessary.

Example 5 Answer:about 50° Find a Missing Angle Measure Because the measures of the side opposite  and the hypotenuse are given, use the sine function. Sine function opp = 12 and hyp = 15.7 ≈ 50° Definition of inverse sine

Example 5 Use a trigonometric function to find the measure of θ. Round to the nearest degree, if necessary. A.32° B.40° C.50° D.58°

Example 6 SKIING The chair lift at a ski resort rises at an angle of 20.75° while traveling up the side of a mountain and attains a vertical height of 1200 feet when it reaches the top. How far does the chair lift travel up the side of the mountain? Use an Angle of Elevation

Example 6 Because the measure of an angle and the length of the opposite side are given in the problem, you can use the sine function to find d. Answer: about 3387 ft Use an Angle of Elevation Sine function Use a calculator. Divide each side by sin o. Multiply each side by d. θ = o, opp = 1200, and hyp = d

Example 6 AIRPLANE A person on an airplane looks down at a point on the ground at an angle of depression of 15°. The plane is flying at an altitude of 10,000 feet. How far is the person from the point on the ground to the nearest foot? A.2588 ft B.10,353 ft C.37,321 ft D.38,637 ft

Example 7 Use Two Angles of Elevation or Depression SIGHTSEEING A sightseer on vacation looks down into a deep canyon using binoculars. The angles of depression to the far bank and near bank of the river below are 61° and 63°, respectively. If the canyon is 1250 feet deep, how wide is the river?

Example 7 Use Two Angles of Elevation or Depression Draw a diagram to model this situation. Because the angle of elevation from a bank to the top of the canyon is congruent to the angle of depression from the canyon to that bank, you can label the angles of elevation as shown. Label the horizontal distance from the near bank to the base of the canyon as x and the width of the river as y.

Example 7 Use Two Angles of Elevation or Depression Tangent function Divide each side by tan 63 o. Multiply each side by x. θ = 63 o, opp = 1250, adj = x For the smaller triangle, you can use the tangent function to find x.

Example 7 Use Two Angles of Elevation or Depression For the larger triangle, you can use the tangent function to find x + y. Tangent function Divide each side by tan 61 o. Multiply each side by x + y. θ = 61 o, opp = 1250, adj = x + y

Example 7 Use Two Angles of Elevation or Depression Answer: about 56 ft Use a calculator. Therefore, the river is about 56 feet wide. Substitute Subtract from each side.

Example 7 HIKING The angle of elevation from a hiker to the top of a mountain is 25 o. After the hiker walks 1000 feet closer to the mountain the angle of elevation is 28 o. How tall is the mountain? A.3791 ft B.4294 ft C.7130 ft D.8970 ft

Example 8 Find f and h using trigonometric functions. Solve a Right Triangle Use a calculator. A.Solve ΔFGH. Round side lengths to the nearest tenth and angle measures to the nearest degree. Substitute. Multiply.

Example 8 Because the measures of two angles are given, H can be found by subtracting F from 90 o. 41.4° + H= 90°Angles H and F are complementary. H≈ 48.6°Subtract. Therefore, H ≈ 49°, f ≈ 18.5, and h ≈ Solve a Right Triangle Answer:H ≈ 49°, f ≈ 18.5, h ≈ 21.0 Substitute. Multiply. Use a calculator.

Example 8 Solve a Right Triangle Because two side lengths are given, you can use the Pythagorean Theorem to find that a = or about You can find B by using any of the trigonometric functions. B.Solve ΔABC. Round side lengths to the nearest tenth and angle measures to the nearest degree.

Example 8 Because B is now known, you can find C by subtracting B from 90 o. 29° + C = 90°Angles B and C are complementary. C= 61° Subtract. Solve a Right Triangle Substitute. Definition of inverse tangent Therefore, B ≈ 29°, C ≈ 61°, and a ≈ Answer:a = 10.3, B ≈ 29°, C ≈ 61° Use a calculator. B ≈ 29°

Example 8 A.a ≈ 44.9, b ≈ 82.7, A = 36° B.a ≈ 40.3, b ≈ 82.7, A = 26° C.a ≈ 40.3, b ≈ 85.4, A = 26° D.a ≈ 54.1, b ≈ 74.4, A = 36° Solve ΔABC. Round side lengths to the nearest tenth and angle measures to the nearest degree.

End of the Lesson