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Evaluating trigonometric functions
Remark. Throughout this document, remember the angle measurement conven- tion, which states that if the measurement of an angle appears without units, then it is assumed to be measured in radians.
Contents
1 Acute and square angles 1
2 Larger angles — the geometric method 2
3 Larger angles — the formulas method 5
1 Acute and square angles
You will need to memorize the values of sine and cosine at the following acute angles: π 6
π 4
= 45◦, and
π 3
= 60◦. You will also need to know their values at the square
angles 0 = 0◦,
π 2
= 90◦, π = 180◦,
3 π 2
= 270◦, and 2π = 360◦.
θ 0
π 6
π 4
π 3
π 2
π
3 π 2
2 π
sin θ 0
cos θ 1
The sine and cosine of the square angles are easy to figure out from the definition of cosine and sine as the x and y coordinates of points on the unit circle. (See Figure P. on page 45 of the textbook.) The sine and cosine of the acute angles listed above can be found by studying a 30◦-60◦-90◦^ triangle and a 45◦-45◦-90◦^ triangle. (See Examples 3 and 4 on page 47 of the textbook.)
Values of the other trigonometric functions at the angles listed above can be found easily, since the other functions are all built from sine and cosine.
Example 1. Question.
Evaluate tan
π 3
and sec
π 4
Answer.
Since tan θ =
sin θ cos θ
, we have
tan
π 3
sin π 3 cos π 3
Similarly,
sec
π 4
cos π 4
2 Larger angles — the geometric method
The first thing to notice is that since sine and cosine repeat their values every 2π radians, if you are asked to evaluate one of these functions at an angle which is not between 0 and 2π, you can simply add or subtract 2π until the angle does lie in that interval. For example,
sin
10 π 3
= sin
10 π 3
− 2 π
= sin
10 π 3
6 π 3
= sin
4 π 3
The next thing to notice is that the value of sine or cosine at any angle θ with 0 ≤ θ ≤ 2 π is essentially determined by their values at an a particular acute angle related to θ, called the reference angle of θ.
Definition 2.1. The reference angle of a non-square angle θ is the acute angle formed between the x-axis and the ray from the origin making an angle of θ with the positive x-axis ray.
Recall that the xy-plane is traditionally divided into four quadrants, hence dividing the non-square angles in one rotation into four types.
Figure 1: The four points on the unit circle with reference angle θ. Their coordinates are A = (cos(θ), sin(θ)), B = (cos(π − θ), sin(π − θ)), C = (cos(π + θ), sin(π + θ)), and D = (cos(2π − θ), sin(2π − θ)). Notice that these four triangles are congruent to each other.
Figure 2: The four Quadrants can be labeled with “All Students Take Calculus.” These labels specify which of sine, cosine, and tangent are positive in that Quadrant.
Example 2. Question.
Evaluate sin
5 π 3
and tan
5 π 4
Answer.
The angle
5 π 3
is in Quadrant IV, so its reference angle is
2 π −
5 π 3
6 π 3
5 π 3
π 3
Sine is negative in Quadrant IV, and sin
π 3
, so sin
5 π 3
The angle
5 π 4
is in Quadrant III, where tangent is positive. Its reference angle is
5 π 4
− π =
5 π 4
4 π 4
π 4
Since tan
π 4
sin π 4 cos π 4
= 1, we have tan
5 π 4
3 Larger angles — the formulas method
The second method for evaluating the sine and cosine of larger angles relies on the following useful trigonometric formulas.
cos(−θ) = cos θ sin(−θ) = − sin θ cos(π − θ) = − cos θ sin(π − θ) = sin θ
Each of these formulas can be derived from geometric properties of the graphs of sine and cosine, or else from those of their definition in terms of the unit circle. For instance, the formula sin(−θ) = −θ expresses the fact that the y-coordinate of the point on the unit circle corresponding to the angle −θ is the negative of the y-coordinate of the point corresponding to the angle θ. Since cos(π − θ) = cos (−(π − θ)) = cos(θ − π), the formula cos(θ − π) = cos(π − θ) = − cos θ
can be interpreted as saying that if you move the cosine curve to the right by π, you get exactly an upside-down cosine curve.
As in the first method, notice that any angle greater than 2π or less than 0 can be made into an equivalent angle by adding or subtracting 2π some number of times. At this point, we can use the formulas given above to reduce any given problem to that of evaluating sine or cosine at an acute angle.
