a. Find the exact value of

Question

sin (theta - pi/6)

b. Find the exact value of
cos (theta - pi/4)

c. For what numbers x, -2pi less than or equal to x less than or equal to 2pi does cos x = -1

d. What is the range of the cosine function?

e. For what numbers x, -pi less than or equal to x less than or equal to pi is the graph of y = cos x increasing?

f. Given: tan alpha = -4/3,
pi/2 < alpha , pi and
cos beta = 1/2, 3pi/2 < beta , 2 pi.
Find: tan (alpha-beta)

Answer 1

a.

sin(-theta)= -sin(theta)

sin(-pi/6)= - sin(pi/6)= -1/2

b.

cos(-theta)=cos(theta)

cos(-pi/6)=cos(pi/6)= sqroot(3)/2

c.

-pi

d.

The cosine function, like all of the trigonometric functions, is periodic about the rotation around a circle. Since the cosine is defined as the adjacent/hypotenuse of a right triangle, you can clearly see that its value can never be greater than one or less than -1 since the hypotenuse is always longer than the adjacent side. It turns out that, indeed, the cosine's range is from -1 to 1, written [-1,1].

e.

-pi/2,pi