# 1 Objectives Define and apply sine, cosine, and tangent in right triangles.

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1 Objectives Define and apply sine, cosine, and tangent in right triangles

2 Trigonometric Ratios for Right Triangles By the AA Similarity Postulate, a right triangle with a given acute angle is similar to every other right triangle with that same acute angle measure. So ∆ABC ~ ∆DEF ~ ∆XYZ, and. These are trigonometric ratios. A trigonometric ratio is a ratio of two sides of a right triangle.

3 Sine, Cosine, Tangent Definitions

4 Story of Chief Soh Cah Toa A young Indian, frustrated by his inability to understand the geometric constructions of his tribe's battle dress, kicked out in anger against a stone and crushed his big toe. Fortunately, he learned from this experience, and began to use study and concentration to solve his problems rather than violence and injure himself. This was especially effective in his study of math, and he went on to become the wisest man of his tribe. He studied many aspects of trigonometry; and even today we remember many of the functions by his name. When he became an adult, the tribal priest gave him a name that reflected his special nature -- one that reminded them of his great discoveries and of the event which changed his life. Because he was troubled throughout his life by the problematic toe that he injured by kicking the stone, he was constantly at the edge of the river, soaking his toe in the cooling waters. For that behavior, he was named Chief Soh Cah Toa.

5 Ex. 1 Sin A = ? 1.Mark angle A. 2.Label the sides in relation to angle A (opp, adj, hyp) 3.Circle the sides that we use for sin 4.Decide which side goes on top of the fraction 5.Write the fraction 13 B C A 5 12 opp adj hyp Sin A = 5/13

6 Ex. 2 Cos B = ? 1.Mark angle B. 2.Label the sides in relation to angle B (opp, adj, hyp) 3.Circle the sides that we use for cos 4.Decide which side goes on top of the fraction 5.Write the fraction 13 B C A 5 12 opp adj hyp Cos B = 5/13

7 Trigonometry with a Calculator Ex. 1 x = ? 1.Mark angle A. 2.Label the sides in relation to angle A (opp, adj, hyp) 3.Decide which trig. func. You can use. 4.Decide which side goes on top of the fraction 5.Write the fraction 6.Solve for x. 13 B C A x opp adj hyp Sin 34 = x/13 x = 13 * sin 34 x = 7.3 34°

8 Trigonometry with a Calculator Ex. 4 x = ? 1.Mark angle A. 2.Label the sides in relation to angle A (opp, adj, hyp) 3.Decide which trig. func. You can use. 4.Decide which side goes on top of the fraction 5.Write the fraction 6.Solve for x. x B C A 20 opp adj hyp Sin 52 = 20/x x = 20/sin 52 x = 25.4 52°

9 Inverse Trigonometric Functions 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.

10 Finding an angle using Trig. Ex. 1 x = ? 1.Mark angle A. 2.Label the sides in relation to angle A (opp, adj, hyp) 3.Decide which trig. func. You can use. 4.Decide which side goes on top of the fraction 5.Write the fraction 6.Solve for x. 14 B C A 5 opp adj hyp Sin x = 5/14 x = sin -1 (5/14) x = 21° x

11 Finding an angle using Trig. Ex. 2 x = ? 1.Mark angle A. 2.Label the sides in relation to angle A (opp, adj, hyp) 3.Decide which trig. func. You can use. 4.Decide which side goes on top of the fraction 5.Write the fraction 6.Solve for x. 14 B C A 5 opp adj hyp Tan x = 5/14 x = tan -1 (5/14) x = 20° x

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