MATH 1A: REVIEW OF TRIGONOMETRIC FUNCTIONS

JOHN VOIGHT

Exercise. How many degrees is the same as π radians? Solution. Since 360^{∘} = 2π, we see that 180^{∘} = π. Exercise. Is it true that cos 30^{∘} = cos 30? Solution. No! Really not! Exercise. Fill in the following table. Draw and label these angles on the unit circle.

Solution.

degrees 0^{∘} 30^{∘} 45^{∘} 120^{∘} 270^{∘} 210^{∘} 150^{∘}
radians 0 π/6 π/4 2π/3 3π/2 7π/6 5π/6

Exercise. Recall that π = 3.14159.... Approximately how many degrees is 1 radian? Solution. Therefore 1 radian is just shy of 1/3 of a half-arc, so it is just slightly less than 60^{∘}. Exercise. Label each quadrant indicating whether the functions cos x and sin x are positive or negative. Solution.

Quadrant cos x sin x
I  +  +
II  −  +
III  −  −
IV  +  −

Exercise. Fill in the following table.

Date: February 4, 2004.

Solution. \begin{matrix}x & 0 & π/3 & π/4 & π/6 & π/2 & π & 3π/2 & −π/4 & 17π/6 \\ cos x & 1 & 1/2 & \sqrt{2}/2 & \sqrt{3}/2 & 0 & −1 & 0 & \sqrt{2}/2 & −\sqrt{3}/2 \\ sin x & 0 & \sqrt{3}/2 & \sqrt{2}/2 & 1/2 & 1 & 0 & −1 & −\sqrt{2}/2 & 1/2 \end{matrix}

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?

Solution.

These graphs have period 2π = 6.2..., which comes from the fact that after 2π radians we have “gone a full circle”, so the values of sin and cos repeat.

The reason that these functions are entirely contained in the horizontal strip [−1, 1] is because they are the x and y coordinates along the unit circle.

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.]

Solution. The function tan x is undefined whenever cos x = 0, i.e. when x = …, −π/2, π/2, 3π/2, 5π/2, . . .. The function cot x is undefined whenever sin x = 0, i.e. when x = …, −π, 0, π, 2π, . . ..

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

Solution.

We can almost read from the graph that tan(π/6) = cos(π/6)/ sin(π/6) = \sqrt{3}/3, and similarly tan(3π/4) = −1, tan(7π/6) = \sqrt{3}/3, and tan(−π/3) = \sqrt{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?

Solution. Drawing the picture, we see that as the angle x approaches π/2 from below, the intersection point along the vertical line tends higher and higher upward: this says that

lim_{x→π/2^{−}} tan(x) = +∞.

Similarly, approaching −π/2 from above, we obtain lim_{x→−π/2^{+}} tan(x) = −∞.

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

Solution.

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.

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

Solution.

The vertical asymptotes for tan(x) occur where this function is undefined, x = …, −π/2, π/2, …, and similarly for cot(x) at x = …, −π, 0, π, . . ..

The functions are no longer contained in the vertical strip because we are no longer taking coordinates along the unit circle, but rather their ratios, which can become arbitrarily positive and negative.