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Solving Right Triangles

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.

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Solving Right Triangles

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

  2. 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. 1 side length and 1 acute angle measure -or- 2 side lengths

  3. 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

  4. Solve the right triangle. Round decimal answers to the nearest tenth. = 70 AB BC 70 tan42o cos42o = ABcos 42o 70 tan 42o = BC = 70 70 0.9004 = BC AB 70 cos 42o 70 0.7431 180o = 90o + 42o + m∠B BC 63.0 ≈ = m∠ B 48o AB 94.2 AB GUIDED PRACTICE Example 1 A Find m∠ B by using the Triangle Sum Theorem. 42o 70 48o B Approximate BCby using a tangent ratio. C Approximate ABby using a cosine ratio. ANSWER The angle measures are 42o, 48o, and 90o. The side lengths are 70 feet, about 63.0 feet, and about 94.2 feet.

  5. Solve a right triangle that has a 40o angle and a 20 inch hypotenuse. = YZ 20 XY 20 sin40o = cos40o 20 ● sin 40o = XY 20 ●cos 40o = YZ 20 ● 0.7660 ≈ YZ XY 20 ● 0.6428 ≈ 180o = 90o + 40o + m∠X BC YZ 15.3 ≈ 12.9 ≈ = m∠ X 50o GUIDED PRACTICE Example 2 Find m∠ Xby using the Triangle Sum Theorem. X 50o 20 in Approximate YZby using a sine ratio. 40o Y Z Approximate ABby using a cosine ratio. ANSWER The angle measures are 40o, 50o, and 90o. The side lengths are 12.9 in., about 15.3 in., and 20 in.

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

  7. 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.

  8. 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°

  9. Inverse trig functions: Ex: Use a calculator to approximate the measure of the acute angle. Round to the nearest tenth. 1. tan A = 0.5 2. sin A = 0.35 3. cos A = 0.64 26.6° 20.5° 50.2°

  10. Use an inverse sine and an inverse cosine sinA = 0.87 cosB = 0.15 a. b. a. m∠A b. m∠B EXAMPLE 2 Example 5 Let ∠Aand ∠Bbe acute angles in a right triangle. Use a calculator to approximate the measures of ∠Aand ∠Bto the nearest tenth of a degree. SOLUTION = sin –1 0.87 ≈ 60.5o = cos –1 0.15≈ 81.4o

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

  12. 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 the nearest tenth.

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

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

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

  16. Homework: • Pg 174 (#4-22 even)

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