 # Solving Right Triangles

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

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

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

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

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

Solve the right triangle. Round to the nearest tenth.
Example 3 37° 24.0 18.1

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.

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°

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°

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

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, mT  90° – 29°  61°. , so ST = 5.7 sinR. Since the acute angles of a right triangle are complementary, mT  90° – 29°  61°.

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°

Solve the right triangle. Round decimals to the nearest tenth.
Example 8

Solve the right triangle. Round decimals to the nearest tenth.
Example 9

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

Homework: Pg 174 (#4-22 even)