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MM2G2 Students will define and apply sine, cosine, and tangent ratio to right triangles. How do you solve right triangles? Lesson 5.4 Day 53 Sunday, April 30, 2017 MM2G2b Explain the relationship of the trigonometric ratios of complementary angles. MM2G2c Solve application problems using the trigonometric ratios. MM2G2 Students will define and apply sine, cosine, and tangent ratio to right triangles. 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. MM2G2b Explain the relationship of the trigonometric ratios of complementary angles. MM2G2c Solve application problems using the trigonometric ratios. MM2G2 Students will define and apply sine, cosine, and tangent ratio to right triangles. 1. Find the measure of the missing angle. Round your answer to the nearest degree. MM2G2b Explain the relationship of the trigonometric ratios of complementary angles. MM2G2c Solve application problems using the trigonometric ratios. MM2G2 Students will define and apply sine, cosine, and tangent ratio to right triangles. 2. Find the measure of the missing angle. Round your answer to the nearest degree . MM2G2b Explain the relationship of the trigonometric ratios of complementary angles. MM2G2c Solve application problems using the trigonometric ratios. MM2G2 3. Students will define and apply sine, cosine, and tangent ratio to right triangles. Find the missing side. Round your answer to the nearest tenth. MM2G2b Explain the relationship of the trigonometric ratios of complementary angles. MM2G2c Solve application problems using the trigonometric ratios. MM2G2 4. Students will define and apply sine, cosine, and tangent ratio to right triangles. Find the missing side. Round your answer to the nearest tenth. MM2G2b Explain the relationship of the trigonometric ratios of complementary angles. MM2G2c Solve application problems using the trigonometric ratios. MM2G2 5. Students will define and apply sine, cosine, and tangent ratio to right triangles. Solve the triangle. Round your answers to the nearest tenth. R RT 14.9 S T mR 20.7 o mS 90o mT 70.3o MM2G2b Explain the relationship of the trigonometric ratios of complementary angles. MM2G2c Solve application problems using the trigonometric ratios. MM2G2 6. Students will define and apply sine, cosine, and tangent ratio to right triangles. Solve the triangle. Round your answers to the nearest tenth. N O M MO 21.6 mN 26.1o mM 90o mO 63.9o MM2G2b Explain the relationship of the trigonometric ratios of complementary angles. MM2G2c Solve application problems using the trigonometric ratios. Y 125 Z z 50 X YX » 134.6 mY 21.8 mX 68.2 8. Solve the right triangle. Round decimals to the tenth. (Hint find all missing side lengths and angle measures) P 22 37˚ Q PQ 13.2 R QR 17.6 mQ 90 mP 53 MM2G2 Students will define and apply sine, cosine, and tangent ratio to right triangles. MM2G2b Explain the relationship of the trigonometric ratios of complementary angles. MM2G2c Solve application problems using the trigonometric ratios. Hint: If you know any trig ratio, use your calculator to find the missing angle Y 5 X 13 12 Z 12 sin Y 13 12 sin1 13 mY 67.4 mZ 90 67.4 mZ 22.6 2. Solve the right triangle. Round decimals to the tenth. (Hint find all missing side lengths and angle measures) 2 a +b =c A 2 17 C cos A 15 17 2 2 2 a + 15 = 17 2 a + 225 = 289 2 a = 64 a= 8 15 B 2 15 cos 17 mA 28.1 mB 90 28.1 mB 61.9 1 MM2G2 Students will define and apply sine, cosine, and tangent ratio to right triangles. GUIDED PRACTICE Solve the right triangle. Round decimal answers to the nearest tenth. A Example 5 42o Find m∠ B by using the Triangle Sum Theorem. c 180o = 90o + 42o + m∠ B 70 48o = m∠ B Approximate BC by using a tangent ratio. a tan 42o = 70 63.0 ≈ a Approximate AB by using a cosine ratio. 70 cos 42o = c 94.2 c 48o C a B ANSWER The angle measures are 42o, 48o, and 90o. The side lengths are 70 feet, about 63.0 feet, and about 94.2 feet. MM2G2b Explain the relationship of the trigonometric ratios of complementary angles. MM2G2c Solve application problems using the trigonometric ratios. MM2G2 Students will define and apply sine, cosine, and tangent ratio to right triangles. GUIDED PRACTICE Solve a right triangle that has a 40o angle and a 20 inch hypotenuse. Example 6 X Find m∠ X by using the Triangle Sum Theorem. 180o = 90o + 40o 50o + m∠ X 20 in 50o = m∠ X Approximate YZ by using a sine ratio. XY sin 40o = 20 12.9 ≈ XY Approximate YZ using a cosine ratio. YZ cos 40o = 20 15.3 ≈ YZ 40o Y Z ANSWER The angle measures are 40o, 50o, and 90o. The side lengths are 12.9 in., about 15.3 in., and 20 in. MM2G2b Explain the relationship of the trigonometric ratios of complementary angles. MM2G2c Solve application problems using the trigonometric ratios. MM2G2 Students will define and apply sine, cosine, and tangent ratio to right triangles. Example 7 Solve the right triangle. Round to the nearest tenth. p cos53 30 p 18.1 q sin53 30 q 24.0 mQ 53 mR 90 mP 37° PQ 30 PR 24.0 QR 18.1 MM2G2b Explain the relationship of the trigonometric ratios of complementary angles. MM2G2c Solve application problems using the trigonometric ratios. MM2G2 Students will define and apply sine, cosine, and tangent ratio to right triangles. MM2G2b Explain the relationship of the trigonometric ratios of complementary angles. MM2G2c Solve application problems using the trigonometric ratios. MM2G2 Students will define and apply sine, cosine, and tangent ratio to right triangles. 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. MM2G2b Explain the relationship of the trigonometric ratios of complementary angles. MM2G2c Solve application problems using the trigonometric ratios. MM2G2 Students will define and apply sine, cosine, and tangent ratio to right triangles. EXAMPLE 2 Example 8 Use an inverse sine and an inverse cosine 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 b. m∠ B = cos –1 –1 0.87 ≈ 60.5o o 0.15≈ 81.4 MM2G2b Explain the relationship of the trigonometric ratios of complementary angles. MM2G2c Solve application problems using the trigonometric ratios. MM2G2 Students will define and apply sine, cosine, and tangent ratio to right triangles. Example 6 Solving Right Triangles 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°. , Since the acute angles of a right triangle are complementary, mT 90° – 29° 61°. ST sin 29 5.7 o ST 5.7 sin 29 ST 2.76 o MM2G2b Explain the relationship of the trigonometric ratios of complementary angles. MM2G2c Solve application problems using the trigonometric ratios. MM2G2 Students will define and apply sine, cosine, and tangent ratio to right triangles. Example 7 Solve the right triangle. Round decimals the nearest tenth. Use Pythagorean Theorem to find c… c 2 22 3 2 c 3.6 Use an inverse trig function to find a missing acute angle… 3 mA tan ( ) 56.3 2 1 Use Triangle Sum Theorem to find the other acute angle… mB 90 56.3 33.7 AB 3.6 BC 3 AC 2 mA 56.3° mB 33.7° mC 90 MM2G2b Explain the relationship of the trigonometric ratios of complementary angles. MM2G2c Solve application problems using the trigonometric ratios. MM2G2 Students will define and apply sine, cosine, and tangent ratio to right triangles. Example 8 PN 2 112 182 PN 21.1 11 mN tan ( ) 31.4 18 1 mP 90 31.4 58.6 MM2G2b Explain the relationship of the trigonometric ratios of complementary angles. MM2G2c Solve application problems using the trigonometric ratios. MM2G2 Students will define and apply sine, cosine, and tangent ratio to right triangles. Example 9 232 TU 2 72 TU 21.9 7 mS cos ( ) 72.3 23 1 mU 90 72.3 17.7 MM2G2b Explain the relationship of the trigonometric ratios of complementary angles. MM2G2c Solve application problems using the trigonometric ratios. MM2G2 Students will define and apply sine, cosine, and tangent ratio to right triangles. tan 55o 55o 555 g g 388.6 ft. 555 555 ft 55o g 55o g MM2G2b Explain the relationship of the trigonometric ratios of complementary angles. MM2G2c Solve application problems using the trigonometric ratios. MM2G2 Students will define and apply sine, cosine, and tangent ratio to right triangles. Homework: ◦ Pg 174 (#4-22 even) MM2G2b Explain the relationship of the trigonometric ratios of complementary angles. MM2G2c Solve application problems using the trigonometric ratios. MM2G2 Students will define and apply sine, cosine, and tangent ratio to right triangles. Ladder Problems http://www.geogebra.org/en/examples/ladder_wall/ladder_wall.html MM2G2b Explain the relationship of the trigonometric ratios of complementary angles. MM2G2c Solve application problems using the trigonometric ratios.