MATH 1A: REVIEW OF TRIGONOMETRIC FUNCTIONS

JOHN VOIGHT

For more practice on the material in this section, please see Appendix D in your text.

Consider the unit circle. The meaning of the functions cos x and sin x are the “horizontal” and “vertical” displacement of the second arm with an angle of measure x in radians.

Remember that the angle increases in the counterclockwise direction, and we measure these angles in radians, not degrees. The angle given by a complete revolution is 360^{∘}, which is the same as 2π radians.

Exercise. How many degrees is the same as π radians?

Calculators usually work in radians and problems (if nothing is specified) are meant to be in radians. A lot of errors can be avoided by not plugging in degrees for x by mistake!

Exercise. Is it true that cos 30^{∘} = cos 30?

Exercise. Fill in the following table:

degrees 0^{∘} 30^{∘} 45^{∘}  150^{∘}
radians 0  2π/3 3π/2 7π/6

Draw and label these angles on the unit circle:

Date: February 4, 2004.

2 JOHN VOIGHT

Exercise. Recall that π = 3.14159.... Approximately how many degrees is 1 radian? Exercise. Label each quadrant indicating whether the functions cos x and sin x are positive or negative.

Quadrant cos x sin x
I + +

II III IV

Exercise. Fill in the following table:

x 0 π/3 π/4 π/6 π/2 π 3π/2 −π/4 17π/6
cos x 1
sin x 0  −\sqrt{2}/2

Exercise. Draw the graphs of sin x and cos x. Use the values tabulated above, if necessary.

What are the periods of these graphs? Where does this come from?

Notice that the functions are entirely contained in the horizontal strip [−1, 1]. Why is this?

We define

tan x = \frac{sin x}{cos x}  and  cot x = \frac{cos x}{sin x}.

Notice that tan x = 1/ cot x, so these two functions are reciprocal functions.

Exercise. For what values of x is tan x undefined? For what values of x is cot x undefined? [Hint: Check when the denominator is zero.]

We can interpret tan x on the unit circle. The function tan x can be measured by using the vertical line through (1, 0): extend the second arm of the angle until it hits the line at a point; the height of this point is the value of tan x.

Exercise. Draw and compute the value of tan x using this procedure for x = π/6, 3π/4, 7π/6, −π/3.

Exercise. What happens to tan x when the angle approaches π/2 from below? (What happens to the point of intersection as the angle is just below π/2 and gets closer and closer?)

What happens when the angle approaches −π/2 from above?

In a similar fashion, we can interpret cot x on the unit circle, measured by using the horizontal line through (0, 1): extend the second arm of the angle until it hits the line at a point, and the displacement of this point is the value of cot x.

Exercise. Label each quadrant indicating whether the functions tan x and cot x are positive or negative.

Quadrant tan x cot x
I  +  +

II

III

IV

Exercise. Draw the graphs of tan x and cot x. Recall the values for which these functions are undefined and how they behave near these values.

1A: REVIEW OF TRIGONOMETRY 5

What are the vertical asymptotes of these graphs? Notice that the functions are not contained in the horizontal strip [−1, 1]. Why is this?