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Understanding Trigonometric Ratios and Inverse Functions in Right Triangles

This lesson focuses on using the Pythagorean Theorem to find missing lengths in right triangles and applying trigonometric ratios to determine angle measures. You will learn how to find sine, cosine, and tangent ratios and utilize calculators for inverse trigonometric calculations. The inverse functions allow for determining angle measures based on known sides of triangles. Step-by-step examples illustrate how to solve for angles and sides while rounding measures accurately. Assignments include practice problems to reinforce learning.

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Understanding Trigonometric Ratios and Inverse Functions in Right Triangles

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  1. 8-4 Trigonometry, day 2 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.

  2. You can use a calculator to find the measure of an angle which is the inverse of the trigonometric ratio (sine, cosine, or tangent of an acute angle). p. 571

  3. Inverse Trigonometric Ratios The expression is read the inverse sine of x and is interpreted as the angle with sine x. Use the thought: If the 30°≈.058, then

  4. ÷ ) ENTER 2nd ( KEYSTROKES: [COS] 13 19 46.82644889 Use a calculator to find the measure of P to the nearest tenth. The measures given are those of the leg adjacent to P and the hypotenuse, so write the equation using the cosine ratio. Answer:So, the measure of P is approximately 46.8°.

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

  6. Solve the right triangle. Round side measures to the nearest hundredth and angle measures to the nearest degree. Step 1 Find mA by using a tangent ratio. Definition of inverse tangent 29.7448813 ≈ mAUse a calculator. So, the measure of A is about 30.

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

  8. So, the measure of ABis about 8.06. 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 ≈ABUse a calculator. Answer:mA ≈ 30, mB ≈ 60, AB ≈ 8.06

  9. 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

  10. 8-4 Assignment day 2 Page 573, 12-15, 36-39, 42-44

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