Presentation on theme: "Solving Right Triangles"— Presentation transcript:
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.To solve a right triangle you need…..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. GUIDED PRACTICEExample 1AFind m∠ B by using the Triangle Sum Theorem.42o180o= 90o + 42o + m∠ B7048o= m∠ B48oBApproximate BC by using a tangent ratio.CApproximate AB by using a cosine ratio.tan 42o=BC 70cos 42o=70 ABANSWER70 tan 42o= BCAB cos 42o=BCABcos 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 ≈BCABAB94.24
5 Solve a right triangle that has a 40o angle and a 20 inch hypotenuse. GUIDED PRACTICEExample 2Find m∠ X by using theTriangle Sum Theorem.X180o= 90o + 40o + m∠ X50o50o= m∠ X20 inApproximate YZ by using a sine ratio.sin 40o=XY2020 ● sin 40o= XY40oY20 ● ≈XYZ12.9 ≈BCApproximate AB by using a cosine ratio.cos 40o=YZ 20ANSWER20 ● cos 40o= YZThe angle measures are 40o, 50o, and 90o. The side lengths are 12.9 in., about 15.3 in., and 20 in.20 ● ≈YZ15.3 ≈YZ5
6 Solve the right triangle. Round to the nearest tenth. Example 337°24.018.1
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 RatiosExample 4Use 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 = sin A = cos A = 0.6426.6°20.5°50.2°
10 Use an inverse sine and an inverse cosine EXAMPLE 2Example 5Let ∠ 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 = 0.87b.cos B = 0.15SOLUTIONa.m ∠ A= sin – ≈60.5ob.m ∠ B= cos – ≈81.4o
11 Solving Right Triangles Example 6Find 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 + ST2Since 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°.
12 Solve the right triangle. Round decimals the nearest tenth. Example 7Use Pythagorean Theorem to find c…3.6Use an inverse trig function to find a missing acute angle…56.3°Use Triangle Sum Theorem to find the other acute angle…33.7°
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.