5-Minute Check 1 Find x and y. A. B. C. D. Starter(s):

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

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

5-Minute Check 4 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 5 A.(–1, 3) B.(–3, 3) C.(5, 3) D.(6, 2) 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).

5-Minute Check 6 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.

Then/Now 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. 8.4 Trigonometry

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

Concept Note: The hypotenuse of a right triangle is never the adjacent leg in these problems. The hypotenuse is just the hypotenuse.

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

Example 1 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 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 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 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 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. 1)

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

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

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

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

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

Example 2 Example 2) 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 CONTINUED Example 2) 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. 2)

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

Example 3 A.1 in. B.11 in. C.16 in. D.15 in. 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? 3)

Concept

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

Example 4 CONTINUED Example 4) 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] nd( ÷)ENTER Answer: So, the measure of  P is approximately 46.8°.

Example 4 A.44.1° B.48.3° C.55.4° D.57.2° Use a calculator to find the measure of  D to the nearest tenth. 4)

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

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

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

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

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

Example 5 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 Solve the right triangle. Round side measures to the nearest tenth and angle measures to the nearest degree. 5)