6/24/2010 ©Evergreen Public Schools Lesson Title Teacher Notes Supplies: scientific calculators for all kids Notes: The goal for this lesson is to help students build on the Exploring Triangles Lab and the Special Right Triangles work with ratios of side lengths. The push is to help students understand that the trig ratios are ratios that compare the side lengths, a concept that many students do not fully understand. Vocabulary: opposite, adjacent, sine, cosine, tangent

©Evergreen Public Schools Learning Target I can find the side lengths of any right triangle using trigonometric ratios. What are some of the ratios about triangles that we’ve already considered in this unit?

©Evergreen Public Schools LaunchLaunch Find all of the missing angles and side lengths of the following right triangle. How are you able to determine them all? 30  8 cm

©Evergreen Public Schools ExploreExplore

5 26  -64  -90  Triangles On your student notes page, sketch a 26  -64  -90  triangle with a short leg of 20 cm. On a 26  -64  -90  triangle, which angle is opposite the shortest side? Which angle is opposite the longest side (the hypotenuse )? 20 cm

©Evergreen Public Schools  -64  -90  Triangles Using the ratios of the side lengths that you found during the Exploring Triangles Lab, find the lengths of the other 2 sides. Check your answers with your partner. How did you find them? Did you and your partner find them the same way? 20 cm 26  64 

©Evergreen Public Schools  -64  -90  Triangles Sketch a 26  -64  -90  triangle with a hypotenuse of 55 cm. Include all of the angle measures and the other two side lengths.

©Evergreen Public Schools  -53  -90  Triangle Sketch a 37  -53  -90  triangle with a long leg of 32 cm. How do you know which angle is opposite the side length of 32 cm? Which angle is adjacent to the side length of 32 cm? 32 cm

©Evergreen Public Schools  -53  -90  Triangle Using the ratios you found for 37  -53  -90  triangles in the Exploring Triangles Lab, find the other two side lengths and label the triangles’ side lengths and angle measures. 32 cm

©Evergreen Public Schools Similar Triangles Triangle 13, 17 and 18 in Exploring Triangles were similar triangles because their angles were all congruent …

©Evergreen Public Schools Similar Triangles and because their sides were proportional.

©Evergreen Public Schools Similar Triangles Mathematicians used those characteristics of similar triangles to make tables of information based on angle measurements. No matter how long the sides are, if the angles are the same, the ratios of the sides will be, as well.

©Evergreen Public Schools Similar Triangles Right triangle trigonometry developed from the very unique relationships among similar triangles.

©Evergreen Public Schools Trigonometric Ratios A trigonometric ratio tells you the ratio of two sides of a right triangle in connection to one of the two non-right angles. The sine of 53  is 0.8 – that means the ratio of the side opposite the 53  angle hypotenuse =  opposite hypotenuse

©Evergreen Public Schools Trigonometric Ratios sin 53  = side opposite the 53  angle hypotenuse How could you use this ratio and the length of the hypotenuse to find the length of the leg opposite the 53º angle? =  opposite 40 cm

©Evergreen Public Schools Trigonometric Ratios The cosine of 53  is the ratio of the side adjacent to the 53  angle hypotenuse What is the value of the cosine of 53  ? How do you know? 24 cm 40 cm 53º

©Evergreen Public Schools Trigonometric Ratios The tangent of 53  is the ratio of the side opposite the 53  angle side adjacent to the 53  angle The tangent of 53  is 1.3. Where do you see the tangent of 53  ? 24 cm 32 cm 53º

©Evergreen Public Schools Trigonometric Ratios On your student notes page, write the following definitions. “A” represents one of the non-right angles in a right triangle. Sine: sin A = side o pposite A s = o h ypotenuse h Cosine: cos A = side a djacent to A c = a h ypotenuse h Tangent: tan A = side o pposite A t = o side a djacent to A a

©Evergreen Public Schools Trigonometric Ratios Use the triangle to find the sine, cosine and tangent of the 37º angle. sin 37º = cos 37º = tan 37º = y cm x cm 40 cm 37º

©Evergreen Public Schools Team Practice Write a trig ratios that could be used to approximate the value of x length to the nearest tenth. Then use a calculator to determine the value. a. b. 42º 8 x 21º 5 x

©Evergreen Public Schools Debrief With your elbow partner, One person describe what the sine of an angle is. The other describe how the sine is similar to and different from the cosine. Be prepared to share what you discussed with the class.

©Evergreen Public Schools Learning Target Did you hit the target? I can find the side lengths of any right triangle using trigonometric ratios. Rate your understanding of the target from 1 to 5. (Not to worry – we’ll do more work with trig ratios!)

©Evergreen Public Schools Practice Practice Sheet Unit 2, 4-1.

©Evergreen Public Schools What is one question that you have about trigonometric ratios?