LESSON 4 1 Right Triangle Trigonometry Targeted TEKS

LESSON 4– 1 Right Triangle Trigonometry

Targeted TEKS P. 4(E) Determine the value of trigonometric ratios of angles and solve problems involving trigonometric ratios in mathematical and real-world problems. P. 4(F) Use trigonometry in mathematical and realworld problems, including directional bearing. Mathematical Processes P. 1(D), P. 1(E)

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

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

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.

Find Values of Trigonometric Ratios Answer:

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

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

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

Use One Trigonometric Value to Find Others Answer:

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

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

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 θ = 35°, adj = x, and hyp = 7 Multiply each side by 7. 5. 73 ≈ x Use a calculator.

Find a Missing Side Length Therefore, x is about 5. 7. Answer: about 5. 7 Check You can check your answer by substituting x = 5. 73 into . x = 5. 73 Simplify.

Find the value of x. Round to the nearest tenth, if necessary. A. 4. 6 B. 8. 1 C. 9. 3 D. 10. 7

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. )

Find a Missing Side Length x = 7891 Use a calculator. So, the competitor must hike about 7891 feet to reach the finish line. Answer: about 7891 ft

WALKING Ernie is walking along the course x, as shown. Find the distance he must walk. A. 569. 7 ft B. 228. 0 ft C. 69. 5 ft D. 8. 5 ft

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

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° Answer: about 50° Definition of inverse sine

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°

Use an Angle of Elevation 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 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. Sine function θ = 20. 75 o, opp = 1200, and hyp = d Multiply each side by d. Divide each side by sin 20. 75 o. Use a calculator. Answer: about 3387 ft

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

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?

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.

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

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

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

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

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

Solve a Right Triangle Substitute. Multiply. Use a calculator. Because the measures of two angles are given, H can be found by subtracting F from 90 o. 41. 4° + H = 90° H ≈ 48. 6° Angles H and F are complementary. Subtract. Therefore, H ≈ 49°, f ≈ 18. 5, and h ≈ 21. 0. Answer: H ≈ 49°, f ≈ 18. 5, h ≈ 21. 0

Solve a Right Triangle B. Solve ΔABC. Round side lengths to the nearest tenth and angle measures to the nearest degree. Because two side lengths are given, you can use the Pythagorean Theorem to find that a = or about 10. 3. You can find B by using any of the trigonometric functions.

Solve a Right Triangle Substitute. Definition of inverse tangent B ≈ 29° Use a calculator. 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. Therefore, B ≈ 29°, C ≈ 61°, and a ≈ 10. 3. Answer: a = 10. 3, B ≈ 29°, C ≈ 61°

Solve ΔABC. Round side lengths to the nearest tenth and angle measures to the nearest degree. 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°

LESSON 4– 1 Right Triangle Trigonometry