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Relating angles and their functions

When working with right triangles, the same rules apply regardless of the orientation of the triangle. In fact, we can evaluate the six trigonometric functions of either of the two acute angles in the triangle in [link] . The side opposite one acute angle is the side adjacent to the other acute angle, and vice versa.

Right triangle with angles alpha and beta. Sides are labeled hypotenuse, adjacent to alpha/opposite to beta, and adjacent to beta/opposite alpha.
The side adjacent to one angle is opposite the other.

We will be asked to find all six trigonometric functions for a given angle in a triangle. Our strategy is to find the sine, cosine, and tangent of the angles first. Then, we can find the other trigonometric functions easily because we know that the reciprocal of sine is cosecant, the reciprocal of cosine is secant, and the reciprocal of tangent is cotangent.

Given the side lengths of a right triangle, evaluate the six trigonometric functions of one of the acute angles.

  1. If needed, draw the right triangle and label the angle provided.
  2. Identify the angle, the adjacent side, the side opposite the angle, and the hypotenuse of the right triangle.
  3. Find the required function:
    • sine as the ratio of the opposite side to the hypotenuse
    • cosine as the ratio of the adjacent side to the hypotenuse
    • tangent as the ratio of the opposite side to the adjacent side
    • secant as the ratio of the hypotenuse to the adjacent side
    • cosecant as the ratio of the hypotenuse to the opposite side
    • cotangent as the ratio of the adjacent side to the opposite side

Evaluating trigonometric functions of angles not in standard position

Using the triangle shown in [link] , evaluate sin α , cos α , tan α , sec α , csc α , and cot α .

Right triangle with sides of 3, 4, and 5. Angle alpha is also labeled.
sin α = opposite  α hypotenuse = 4 5 cos α = adjacent to  α hypotenuse = 3 5 tan α = opposite  α adjacent to  α = 4 3 sec α = hypotenuse adjacent to  α = 5 3 csc α = hypotenuse opposite  α = 5 4 cot α = adjacent to  α opposite  α = 3 4

Using the triangle shown in [link] , evaluate sin   t , cos   t , tan   t , sec   t , csc   t , and cot   t .

Right triangle with sides 33, 56, and 65. Angle t is also labeled.

s i n   t = 33 65 , cos   t = 56 65 , t a n   t = 33 56 , sec   t = 65 56 , csc   t = 65 33 , cot   t = 56 33

Finding trigonometric functions of special angles using side lengths

We have already discussed the trigonometric functions as they relate to the special angles on the unit circle. Now, we can use those relationships to evaluate triangles that contain those special angles. We do this because when we evaluate the special angles in trigonometric functions, they have relatively friendly values, values that contain either no or just one square root in the ratio. Therefore, these are the angles often used in math and science problems. We will use multiples of 30° , 60° , and 45° , however, remember that when dealing with right triangles, we are limited to angles between  and 90° .

Suppose we have a 30° , 60° , 9 triangle, which can also be described as a π 6 ,   π 3 , π 2 triangle. The sides have lengths in the relation s , 3 s , 2 s . The sides of a 45° , 45° , 90° triangle, which can also be described as a π 4 , π 4 , π 2 triangle, have lengths in the relation s , s , 2 s . These relations are shown in [link] .

Two side by side graphs of circles with inscribed angles. First circle has angle of pi/3 inscribed. Second circle has angle of pi/4 inscribed.
Side lengths of special triangles

We can then use the ratios of the side lengths to evaluate trigonometric functions of special angles.

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Source:  OpenStax, Contemporary math applications. OpenStax CNX. Dec 15, 2014 Download for free at http://legacy.cnx.org/content/col11559/1.6
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