LESSON 8–4 Trigonometry

Five-Minute Check (over Lesson 8–3) TEKS Then/Now New Vocabulary Key Concept: Trigonometric Ratios Example 1: Find Sine, Cosine, and Tangent Ratios Example 2: Use Special Right Triangles to Find Trigonometric Ratios Example 3: Real-World Example: Estimate Measures Using Trigonometry Key Concept: Inverse Trigonometric Ratios Example 4: Find Angle Measures Using Inverse Trigonometric Ratios Example 5: Solve a Right Triangle Lesson Menu

Find x and y. A. B. C. D. 5-Minute Check 1

Find x and y. A. x = 5, y = 5 B. x = 5, y = 45 C. D. 5-Minute Check 2

The length of the diagonal of a square is centimeters The length of the diagonal of a square is centimeters. Find the perimeter of the square. A. 15 cm B. 30 cm C. 45 cm D. 60 cm 5-Minute Check 3

The side of an equilateral triangle measures 21 inches The side of an equilateral triangle measures 21 inches. Find the length of an altitude of the triangle. A. in. B. 12 in. C. 14 in. D. in. 5-Minute Check 4

ΔMNP is a 45°-45°-90° triangle with right angle P ΔMNP is a 45°-45°-90° triangle with right angle P. Find the coordinates of M in Quadrant II for P(2, 3) and N(2, 8). A. (–1, 3) B. (–3, 3) C. (5, 3) D. (6, 2) 5-Minute Check 5

The hypotenuse of a 30°-60°-90° triangle measures inches The hypotenuse of a 30°-60°-90° triangle measures inches. What is the length of the side opposite the 30° angle? A. 10 in. B. 20 in. C. D. 5-Minute Check 6

Mathematical Processes G.1(C), G.1(G) Targeted TEKS G.9(A) Determine the lengths of sides and measures of angles in a right triangle by applying the trigonometric ratios sine, cosine, and tangent to solve problems. Mathematical Processes G.1(C), G.1(G) TEKS

Find trigonometric ratios using right triangles. You used the Pythagorean Theorem to find missing lengths in right triangles. Find trigonometric ratios using right triangles. Use trigonometric ratios to find angle measures in right triangles. Then/Now

trigonometry trigonometric ratio sine cosine tangent inverse sine inverse cosine inverse tangent Vocabulary

Concept

Find Sine, Cosine, and Tangent Ratios A. Express sin L as a fraction and as a decimal to the nearest hundredth. Answer: Example 1

Find Sine, Cosine, and Tangent Ratios B. Express cos L as a fraction and as a decimal to the nearest hundredth. Answer: Example 1

Find Sine, Cosine, and Tangent Ratios C. Express tan L as a fraction and as a decimal to the nearest hundredth. Answer: Example 1

Find Sine, Cosine, and Tangent Ratios D. Express sin N as a fraction and as a decimal to the nearest hundredth. Answer: Example 1

Find Sine, Cosine, and Tangent Ratios E. Express cos N as a fraction and as a decimal to the nearest hundredth. Answer: Example 1

Find Sine, Cosine, and Tangent Ratios F. Express tan N as a fraction and as a decimal to the nearest hundredth. Answer: Example 1

A. Find sin A. A. B. C. D. Example 1

B. Find cos A. A. B. C. D. Example 1

C. Find tan A. A. B. C. D. Example 1

D. Find sin B. A. B. C. D. Example 1

E. Find cos B. A. B. C. D. Example 1

F. Find tan B. A. B. C. D. Example 1

The side adjacent to the 60° angle has a measure of x. Use Special Right Triangles to Find Trigonometric Ratios Use a special right triangle to express the cosine of 60° as a fraction and as a decimal to the nearest hundredth. Draw and label the side lengths of a 30°-60°-90° right triangle, with x as the length of the shorter leg and 2x as the length of the hypotenuse. The side adjacent to the 60° angle has a measure of x. Example 2

Definition of cosine ratio Use Special Right Triangles to Find Trigonometric Ratios Definition of cosine ratio Substitution Simplify. Example 2

Use a special right triangle to express the tangent of 60° as a fraction and as a decimal to the nearest hundredth. A. B. C. D. Example 2

Estimate Measures Using Trigonometry EXERCISING A fitness trainer sets the incline on a treadmill to 7°. The walking surface is 5 feet long. Approximately how many inches did the trainer raise the end of the treadmill from the floor? Let y be the height of the treadmill from the floor in inches. The length of the treadmill is 5 feet, or 60 inches. Example 3

Use a calculator to find y. Estimate Measures Using Trigonometry Multiply each side by 60. Use a calculator to find y. KEYSTROKES: 60 7 7.312160604 ENTER SIN Answer: The treadmill is about 7.3 inches high. Example 3

CONSTRUCTION The bottom of a handicap ramp is 15 feet from the entrance of a building. If the angle of the ramp is about 4.8°, about how high does the ramp rise off the ground to the nearest inch? A. 1 in. B. 11 in. C. 16 in. D. 15 in. Example 3

Concept

Use a calculator to find the measure of P to the nearest tenth. Find Angle Measures Using Inverse Trigonometric Ratios Use a calculator to find the measure of P to the nearest tenth. Example 4

Answer: So, the measure of P is approximately 46.8°. Find Angle Measures Using Inverse Trigonometric Ratios The measures given are those of the leg adjacent to P and the hypotenuse, so write the equation using the cosine ratio. KEYSTROKES: [COS] 13 19 46.82644889 2nd ( ÷ ) ENTER Answer: So, the measure of P is approximately 46.8°. Example 4

Use a calculator to find the measure of D to the nearest tenth. B. 48.3° C. 55.4° D. 57.2° Example 4

Solve a Right Triangle Solve the right triangle. Round side measures to the nearest hundredth and angle measures to the nearest degree. Example 5

Step 1 Find mA by using a tangent ratio. Solve a Right Triangle Step 1 Find mA by using a tangent ratio. Definition of inverse tangent 29.7448813 ≈ mA Use a calculator. So, the measure of A is about 30. Example 5

Step 2 Find mB using complementary angles. Solve a Right Triangle Step 2 Find mB using complementary angles. mA + mB = 90 Definition of complementary angles 30 + mB ≈ 90 mA ≈ 30 mB ≈ 60 Subtract 30 from each side. So, the measure of B is about 60. Example 5

Step 3 Find AB by using the Pythagorean Theorem. Solve a Right Triangle Step 3 Find AB by using the Pythagorean Theorem. (AC)2 + (BC)2 = (AB)2 Pythagorean Theorem 72 + 42 = (AB)2 Substitution 65 = (AB)2 Simplify. Take the positive square root of each side. 8.06 ≈ AB Use a calculator. Example 5

So, the measure of AB is about 8.06. Solve a Right Triangle So, the measure of AB is about 8.06. Answer: mA ≈ 30, mB ≈ 60, AB ≈ 8.06 Example 5

Solve the right triangle Solve the right triangle. Round side measures to the nearest tenth and angle measures to the nearest degree. A. mA = 36°, mB = 54°, AB = 13.6 B. mA = 54°, mB = 36°, AB = 13.6 C. mA = 36°, mB = 54°, AB = 16.3 D. mA = 54°, mB = 36°, AB = 16.3 Example 5

LESSON 8–4 Trigonometry