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Solving Right Triangles: Methods and Examples

jamesperry avatar
jamesperry
26 diapositives · 353 Views · Jan 05, 2020

Learn how to find missing angles and sides in right triangles through trigonometric ratios and the Pythagorean Theorem. Practice solving various examples. Discover how to apply inverse trigonometric functions to determine angle measures accurately.

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  1. Solving Right Triangles How do you solve right triangles? Day 53 Lesson 5.4 Sunday, January 5, 2020

  2. 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.

  3. Find the measure of the missing angle. • Round your answer to the nearest degree.

  4. Find the measure of the missing angle. • Round your answer to the nearest degree .

  5. 3. Find the missing side. Round your answer to the nearest tenth.

  6. 4. Find the missing side. Round your answer to the nearest tenth.

  7. 5. Solve the triangle. Round your answers to the nearest tenth. R S T

  8. 6. Solve the triangle. Round your answers to the nearest tenth. N O M

  9. 7. Solve the right triangle. Round decimals to the tenth.(Hint find all missing side lengths and angle measures) Y z 125 Z X 50

  10. 8. Solve the right triangle. Round decimals to the tenth.(Hint find all missing side lengths and angle measures) P 22 37˚ R Q

  11. Assignment Day 53

  12. 1. Find the missing angle measures. Hint: If you know any trig ratio, use your calculator to find the missing angle Y 13 5 12 Z X

  13. 2. Solve the right triangle. Round decimals to the tenth. (Hint find all missing side lengths and angle measures) A 17 15 C B

  14. = 70 c a 70 tan42o cos42o = 63.0 ≈ 180o = 90o + 42o + m∠B a = m∠ B 48o 94.2 c GUIDED PRACTICE Solve the right triangle. Round decimal answers to the nearest tenth. A Example 5 42o c Find m∠ Bby using the Triangle Sum Theorem. 70 48o Approximate BCby using a tangent ratio. B C a ANSWER Approximate ABby using a cosine ratio. The angle measures are 42o, 48o, and 90o. The side lengths are 70 feet, about 63.0 feet, and about 94.2 feet.

  15. = YZ 20 XY 20 sin40o = cos40o 180o = 90o + 40o + m∠X XY YZ 15.3 ≈ 12.9 ≈ = m∠ X 50o GUIDED PRACTICE Solve a right triangle that has a 40o angle and a 20 inch hypotenuse. Example 6 X Find m∠ Xby using the Triangle Sum Theorem. 50o 20 in Approximate YZby using a sine ratio. 40o Y Z ANSWER Approximate YZ using a cosine ratio. The angle measures are 40o, 50o, and 90o. The side lengths are 12.9 in., about 15.3 in., and 20 in.

  16. Example 7 37° 24.0 18.1 Solve the right triangle. Round to the nearest tenth.

  17. 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.

  18. sinA = 0.87 cosB = 0.15 a. b. a. m∠A b. m∠B EXAMPLE 2 Use an inverse sine and an inverse cosine Example 8 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 81.4o = cos –1 0.15≈

  19. , Solving Right Triangles Example 6 Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree. Method 2: Method 1: By the Pythagorean Theorem, 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°.

  20. 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.

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

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

  23. The Washington Monument is 555 feet high. A person in the top of the monument can see Mr. Zittrouer and calculates the angle of depression to be 55o. How far is Mr. Zittrouer from the base of the monument? 55o 555 555 ft 55o g 55o g

  24. Homework: • Pg 174 (#4-22 even)

  25. Ladder Problems http://www.geogebra.org/en/examples/ladder_wall/ladder_wall.html