14.1 Similar Triangles and the Tangent Ratio Per Date Trigonometric Ratios Investigate the relationship of the tangent ratio.

Transcription

1 14.1 Similar Triangles and the Tangent Ratio Per Date Trigonometric Ratios Investigate the relationship of the tangent ratio. Using the space below, draw at least right triangles, each of which has one acute angle measure of. Since each triangle has a right angle and a congruent acute angle, the triangles are similar by. Page 1

2 14.1 Similar Triangles and the Tangent Ratio Per Date Measure the length of the closet (adjacent) leg and the leg across from (opposite) your designated acute angle of each triangle, then find the ratio of the opposite side to the adjacent side, and record your data in the table on the net page. Length of opposite side (in mm) Length of adjacent side (in mm) length of opposite length of adjacent (to the thousandths) 1. What do you notice? 2. Compare your observation with your group. What did they notice about their ratios? 3. Fill in the table below to show the ratios of each person in your group. Group Members Degree Ratio 4. Use what you discovered to make a conjecture about the ratio of height to length of the base in similar right triangles. In similar right triangles, the ratio of the opposite leg to the adjacent leg. Page 2

3 14.1 Similar Triangles and the Tangent Ratio Per Date Objective: To be able to calculate tangents of acute angles in right triangles and to use tangents to determine side lengths in triangles. Trigonometry: comes from the Greek words meaning. leg opposite " Tangent Ratio (#VOC): tangent of =, leg adjacent " which is abbreviated. (only in right triangles) C Eample 1: Epress the ratio equivalent to tan E and tan F. a) b) c) E F 8 E D 6 F D F D E Eample 2: Find the length of the missing leg. Round your answer to the nearest tenth. tan 62 = 1 First, convert the a) tan 46 = b) tan = = tan 62 into a 1 decimal rounded to 4 decimal places = = 28.2 Then, use the appropriate algebra to solve for. 3 Page 3

4 14.1 Similar Triangles and the Tangent Ratio Per Date Practice: Find the value of. Round lengths of segments to the nearest tenth Find all angles whose tangent ratio equals 3 4. Y Z W C 24 Page 4

5 14.2 Homework Per Date Homework: Epress the ratio equivalent to tan and tan Solve for. Round your answers to the nearest hundredths. 3. tan 34 = 4. tan 2 = 4 20 Find the value of. Round lengths of segments to the nearest tenth Triangle MOP is similar to triangle OS. Find all angles whose tangent ratio equals M P 2 O 6 S 9 Page

6 14.3 The Sine and Cosine Ratios Per Date Objective: To be able to calculate the sine and cosine of acute angles in right triangles and to use sine and cosine to determine unknown measures in triangles. Trigonometric Ratios (only in right triangles) C Review Tangent Ratio: tangent of = leg opposite ", abbreviated leg adjacent " Sine Ratio (#VOC): sine of = leg opposite ", abbreviated hypotenuse Cosine Ratio (#VOC): cosine of = leg adjacent ", abbreviated hypotenuse Eample 1: Epress the ratio equivalent to sin P, cos P, sin Q, and cos Q. a) b) c) P Q 4 3 R R 6 P 14 Q Q 24 P 20 R Sin P = Sin Q = Cos P = Cos Q = Sin P = Sin Q = Cos P = Cos Q = Sin P = Sin Q = Cos P = Cos Q = Page 6

7 14.3 The Sine and Cosine Ratios Per Date Reflect: 1. What do you notice about sin P and Cos Q for each of the problems on the previous page? 2. Look at the first right triangle. Since it is a right triangle,!! = 90. What is the sum of!! and!!? P Q 4 3 R 3. ngles that add up to 90 are called angles. 4.. What is the relationship between Sin P and Cos Q for all the problems on the previous page? Fill in the blanks with the help from your teacher: 6. If angles and are then Sin will always equal Cos. 7. This is where the name Cosine comes from. It is the sine of the co-. For eample, cos = sin, if and only if m + m =. Page 7

8 14.3 The Sine and Cosine Ratios Per Date Practice: 1. Epress the ratio equivalent to sin E and sin F. E 8 10 D 6 F Determine whether each equation is correct. nswer yes or no for each equation. If no, find the correct ratio. 2. sin = 4 4. sin = cos = 3. cos = Find the value of. Round lengths of segments to the nearest tenth Let the sin 47 = Find the angle measure, in degrees, for cos = Let the sin 30 = 1 2. Find the angle measure, in degrees, for cos = 1 2. Page 8

9 14.4 Homework Per Date Homework: Epress the ratio equivalent to sin, sin, cos, and cos Find the value of. Round lengths of segments to the nearest tenth Determine whether each equation is correct. nswer yes or no for each equation. 6. sin = cos = 8. cos = Let the cos 2 = Find the angle measure, in degrees, for sin = Let the sin 60 = 3 2. Find the angle measure, in degrees, for cos = 3 2. Page 9