Sine, Cosine, and Tangent

Introduction: In this lesson, three trigonometric ratios (sine, cosine, and tangent) will be defined and applied. These involve ratios of the lengths of the sides in a right triangle.
In a right triangle, one angle is 90º and the side across from this angle is called the hypotenuse. The two sides which form the 90º angle are called the legs of the right triangle. We show a right triangle below. The legs are defined as either “opposite” or “adjacent” (next to) the angle A.

We shall call the opposite side “opp,”  the adjacent side “adj” and the hypotenuse “hyp.”
Definitions: In the following definitions, sine is called “sin,” cosine is called “cos” and tangent is called “tan.” The origin of these terms relates to arcs and tangents to a circle.
1. sin(A) =
2. cos(A) =
3. tan(A) =
For example, in the triangle below, in relation to angle θ, opp = 5, adj = 6, and by the Pythagorean Theorem, hyp = .

By the three definitions we have:
sin(θ) =
cos(θ) =
tan(θ) =
Often, fractions involving radicals are rewritten (Radical Simplifying) so that there is not a radical expression in the denominator. Then we have sin(θ) = and cos(θ) = .

We can also find the measure of the angle θ when we know any of these three trigonometric ratios. In this case, tan(θ) = . Using a calculator, we can determine what angle has this tangent. Using a TI-83 calculator, we press and get
Let's Practice:
1. In the triangle below, what is the sin(θ) and what is the length of the adjacent side? What is the measure of angle θ?

The adjacent side can be found using the Pythagorean Theorem and is . The sin(θ) = Using we get . Notice that the adjacent side is . If we had used we would still get .
1. In the triangle below, what is the length of side x?

We know that . Therefore, We solve this equation by multiplying by 6 and get x = 2.9088.
1. In the triangle below, what is the length of the hypotenuse x?
 41º 7
Since 7 is the length of the adjacent side, we have cos(41º) = . Therefore, cos(41º) . We solve this equation by multiplying by x and dividing by 0.7547. We get .

Examples
 In a right triangle, the legs are 6 and . What is the length of the hypotenuse and what are the measures of the two non-right (acute) angles? What is your answer?
 In a right triangle, the hypotenuse is 23 and one acute angle is 13º. What are the lengths of the legs? What is your answer?
 In a right triangle, one leg is 9 and the angle opposite this leg is 57º. What are the lengths of the other legs and the hypotenuse? What is your answer?

M Ransom

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