Lesson Title 2: Problem TK Solving with Trigonometric Ratios

Transcription

1 Part UNIT RIGHT solving TRIANGLE equations TRIGONOMETRY and inequalities Lesson Title : Problem TK Solving with Trigonometric Ratios Georgia Performance Standards MMG: Students will define and apply sine, cosine, and tangent ratios to right triangles. c. Solve application problems using the trigonometric ratios. Essential Questions 1. How can trigonometric ratios be used to solve problems (such as finding area and or perimeter) of other geometric shapes?. How can right triangles and trigonometric ratios be used to indirectly measure heights and distances? 3. In what ways can trigonometric ratios be used in engineering and buildings? WORDS TO KNOW adjacent side 166 Unit : Right Triangle Trigonometry the side of a triangle that is formed by one of the sides of the angle angle of elevation the angle formed by an imaginary horizontal line from an observer to the line of sight to an object up from the horizontal from the same observer angle of depression the angle formed by an imaginary horizontal line from an observer to the line of sight to an object down from the horizontal from the same observer cosine the ratio of the length of the side adjacent to the angle to the length of the hypotenuse hypotenuse longest side of a right triangle; the side opposite the 90 angle opposite side the side of a triangle that is NOT formed by one of the sides of the angle sine the ratio of the length of the side opposite to the angle to the length of the hypotenuse tangent the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle

2 Guided Practice..1 An angle of depression is the angle below horizontal that an observer must look to view an object that is below the observer. An angle of elevation is the angle above horizontal that observer must look to view an object that is above the observer. Example 1 An airplane pilot sees a campfire at a 14 angle of depression. The airplane s altitude is,450 meters. What is the airplane s ground distance from the campfire? 1. Start by drawing an appropriate right triangle to model the situation.. The length of the side opposite the 14 angle is given as,450 m. The ground distance is the same as the length of the side adjacent to the 14 angle in the triangle drawn. 3. The length of the opposite side and adjacent side to the 14 angle are given in the problem. Use the tangent ratio: tan 14 =, 450. x 4. Use the table from Lesson.1. (or your calculator) to find an approximate value for tan 14. The tangent of 14 is between 0.68 (tan 15 ) and (tan 10 ), or approximately Cross multiply and solve the equation., = x 0. 49x =, 450 x = 9,

3 Example Three sides of a right triangle measure 3 inches, 4 inches, and 5 inches. What is the measure of the angle opposite the side with length 4 inches? 1. Draw an appropriate right triangle to represent the situation.. Find the ratio that represents the sine of the angle in question. The opposite ratio for sine is. Write the equation: sin. hypotenuse x = 4 5 = Look at the table you created in lesson.1.. Notice that the angle whose sine is 0.8 is between 50 (sin 50 = 0.766) and 55 (sin 55 = 0.819). 4. Interpolate the value of x by dividing the difference between 0.8 and sin 50 by the difference between sin 55 and sin 50, multiplying by 5 and adding the result to This yields = = The measure of x is about Unit : Right Triangle Trigonometry

4 PRaCtiCe UNIt RigHt triangle trigonometry Skills Practice..1: Using Trigonometric Ratios to Solve Triangles For each example, use a trigonometric ratio to find the value of x continued 169

5 PRaCtiCe UNIt RigHt triangle trigonometry Unit : Right Triangle Trigonometry

6 Guided Practice.. If you know the sine of an angle you can use the trig identity sin solve for the cosine of that angle. Example 1 If sin a = 0.4, what is the value of cos a? (Use the identity sin 1. Write the equation for the identity. sin a + cos a = 1 a + cos a = 1 to a + cos a = 1.). Substitute the given value of sin a into the identity. ( 04. ) + cos a = 1 3. Solve for cos a. ( 04. ) + cos a = cos a = 1 cos a = 084. cos a = The value of cos a is Example 4 Given that cos a = , what is the value of sin a? (Use the identity sin a + cos a = 1.) 1. First take the square root cos 4 a of to find cos a. cos a = = 05.. Substitute the value for cos a into the identity and solve to find sin a. sin a = 1 sin a = Take the square root to find sin a. sin a = 075. = The value of sin a is

7 PRACTICE UNIT RIGHT TRIANGLE TRIGONOMETRY Skills Practice..: Using the Pythagorean Identity If cos x = 0.45, find the following quantities. 1. cos x. sin x 3. sin x 4. sin (90 x) 5. cos (90 x) If cos b = s, find the following quantities, in terms of s. 6. cos b 7. sin b 8. sin b 9. sin (90 b) 10. cos (90 b) 17 Unit : Right Triangle Trigonometry