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**Solving Right Triangles**

How do you solve right triangles? M2 Unit 2: Day 6

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**To solve a right triangle you need…..**

Every right triangle has one right angle, two acute angles, one hypotenuse, and two legs. To SOLVE A RIGHT TRIANGLE means to find all 6 parts. To solve a right triangle you need….. 1 side length and 1 acute angle measure -or- 2 side lengths

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**Given one acute angle and one side:**

To find the missing acute angle, use the Triangle Sum Theorem. To find one missing side length, write an equation using a trig function. To find the other side, use another trig function or the Pythagorean Theorem

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**Solve the right triangle. Round decimal answers to the nearest tenth.**

GUIDED PRACTICE Example 1 A Find m∠ B by using the Triangle Sum Theorem. 42o 180o = 90o + 42o + m∠ B 70 48o = m∠ B 48o B Approximate BC by using a tangent ratio. C Approximate AB by using a cosine ratio. tan 42o = BC 70 cos 42o = 70 AB ANSWER 70 tan 42o = BC AB cos 42o = BC AB cos 42o = The angle measures are 42o, 48o, and 90o. The side lengths are 70 feet, about 63.0 feet, and about 94.2 feet. 63.0 ≈ BC AB AB 94.2 4

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**Solve a right triangle that has a 40o angle and a 20 inch hypotenuse.**

GUIDED PRACTICE Example 2 Find m∠ X by using the Triangle Sum Theorem. X 180o = 90o + 40o + m∠ X 50o 50o = m∠ X 20 in Approximate YZ by using a sine ratio. sin 40o = XY 20 20 ● sin 40o = XY 40o Y 20 ● ≈ XY Z 12.9 ≈ BC Approximate AB by using a cosine ratio. cos 40o = YZ 20 ANSWER 20 ● cos 40o = YZ The angle measures are 40o, 50o, and 90o. The side lengths are 12.9 in., about 15.3 in., and 20 in. 20 ● ≈ YZ 15.3 ≈ YZ 5

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**Solve the right triangle. Round to the nearest tenth.**

Example 3 37° 24.0 18.1

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If you know the sine, cosine, or tangent of an acute angle measure, you can use the inverse trigonometric functions to find the measure of the angle.

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**Calculating Angle Measures from**

Trigonometric Ratios Example 4 Use your calculator to find each angle measure to the nearest tenth of a degree. A. cos-1(0.87) B. sin-1(0.85) C. tan-1(0.71) cos-1(0.87) 29.5° sin-1(0.85) 58.2° tan-1(0.71) 35.4°

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**Inverse trig functions:**

Ex: Use a calculator to approximate the measure of the acute angle. Round to the nearest tenth. 1. tan A = sin A = cos A = 0.64 26.6° 20.5° 50.2°

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**Use an inverse sine and an inverse cosine**

EXAMPLE 2 Example 5 Let ∠ A and ∠ B be acute angles in a right triangle. Use a calculator to approximate the measures of ∠ A and ∠ B to the nearest tenth of a degree. a. sin A = 0.87 b. cos B = 0.15 SOLUTION a. m ∠ A = sin – ≈ 60.5o b. m ∠ B = cos – ≈ 81.4o

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**Solving Right Triangles**

Example 6 Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree. Method 1: By the Pythagorean Theorem, Method 2: RT2 = RS2 + ST2 (5.7)2 = 52 + ST2 Since the acute angles of a right triangle are complementary, mT 90° – 29° 61°. , so ST = 5.7 sinR. Since the acute angles of a right triangle are complementary, mT 90° – 29° 61°.

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**Solve the right triangle. Round decimals the nearest tenth.**

Example 7 Use Pythagorean Theorem to find c… 3.6 Use an inverse trig function to find a missing acute angle… 56.3° Use Triangle Sum Theorem to find the other acute angle… 33.7°

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**Solve the right triangle. Round decimals to the nearest tenth.**

Example 8

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**Solve the right triangle. Round decimals to the nearest tenth.**

Example 9

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**Solve the right triangle. Round decimals to the nearest tenth.**

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Homework: Pg 174 (#4-22 even)

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