Class Greeting
Chapter 8 – Lesson 4 Trigonometric Ratios
Objective Use trigonometric ratios to find angle measures in right triangles and to solve real-world problems.
Using given measures to find the unknown angle measures or side lengths of a triangle is known as solving a triangle. To solve a right triangle, you need to know two side lengths or one side length and an acute angle measure. We can use trigonometric ratios to solve right triangles.
Basic Trigonometric Ratios SOH-CAH-TOA
Example 1: Trigonometric Ratios Find the Trigonometric Ratios of each angle.
Find sin A, cos A, tan A, sin B, cos B, and tan B. Example 4-1f
Answer: Example 4-1g
Hypotenuse = 2(shorter leg) Remember this from 8.3? Find the values of x and y. Give your answers in simplest radical form. Hypotenuse = 2(shorter leg) 22 = 2x 11 = x Now we can find the sin 30° by using our trigonometric functions. Now use your calculator to find the sin 30°. KEYSTROKES: 30 0.5 SIN ENTER Therefore the sin 30° = 0.5.
Since the sin 30° = 0.5. Conversely… if we know that the sin 1 = 0.5, what is the measure of 1? m 1 = 30° This is written as sin-1(0.5) = 30°. (Try it on your calculator) Finds the ratio of the sides. Finds the measure of the angle.
Example 2: Calculating Angle Measures from Trigonometric Ratios Use your calculator to find each angle measure to the nearest degree. A. cos-1(0.87) B. sin-1(0.85) C. tan-1(0.71) cos-1(0.87) 30° sin-1(0.85) 58° tan-1(0.71) 35°
Now we can use our calculator to find the angle measures. Remember this? Find sin A, cos A, tan A, sin B, cos B, and tan B. Now we can use our calculator to find the angle measures. Example 4-1f
Do these answers make sense? m B 53° Remember your answers? Answer: Now use your calculator to find the angle measures. m A 37° Do these answers make sense? m B 53° Example 4-1g
Do these answers make sense? Find sin A, cos A, tan A, sin B, cos B, and tan B. 53° 37° The answers make sense, A and B are complementary. Example 4-1f
Example 3: Solving a Right in ∆ the Coordinate Plane The coordinates of the vertices of ∆PQR are P(–3, 3), Q(2, 3), and R(–3, –4). Find the side lengths to the nearest hundredth and the angle measures to the nearest degree. Step 1 Find the side lengths. Plot points P, Q, and R. PR = 7 PQ = 5 Y P Q X mP = 90° R mR 90° – 54° 36°
Find the unknown measures Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree. c2 = a2 + b2 (5.7)2 = 52 + ST2 Since the acute angles of a right triangle are complementary, mT 90° – 29° 61°.
Example 4 Baldwin St. in Dunedin, New Zealand, is the steepest street in the world. It has a grade of 38%. To the nearest degree, what angle does Baldwin St. make with a horizontal line? A 38% grade means the road rises (or falls) 38 ft for every 100 ft of horizontal distance. 100 ft 38 ft A B C
Use a calculator to find tan to the nearest ten thousandth. Review Use a calculator to find tan to the nearest ten thousandth. KEYSTROKES: 56 1.482560969 TAN ENTER Answer: Example 4-2a
Kahoot!
Lesson Summary: Objective Use trigonometric ratios to find angle measures in right triangles and to solve real-world problems.
Preview of the Next Lesson: Objective Solve problems involving angles of elevation and angles of depression.
Stand Up Please
Lesson Quiz: Part I Use your calculator to find each angle measure to the nearest degree. 1. cos-1 (0.97) 2. tan-1 (2) 3. sin-1 (0.59)
Lesson Quiz: Part II Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree. 4. 5.
Lesson Quiz: Part III 6. The coordinates of the vertices of ∆MNP are M (–3, –2), N(–3, 5), and P(6, 5). Find the side lengths to the nearest hundredth and the angle measures to the nearest degree.