Question

Can you find

(a) a sine graph which touches the lines y = 3 and y = 1?

(b) a cosine graph which crosses the x-axis at x = 1 and x = −1?

(c) a tangent graph which passes through the point (π/3, 0) and for which the line x= π/2 is an asymptote?
Part 2
Write a trigonometric equation that is not an identity. Explain how you know it is not an identity.
Explain why you cannot square each side of the equation when verifying a trigonometric identity.
Explain why both sides of a trigonometric identity are often rewritten in terms of sine and cosine.

Answer 1

(a) the center line is at y = (3+1)/2 = 2
the amplitude is (3-1)/2 = 1
So, y = 2+sinx

(b) 1/2 period is 1-(-1) = 2, so the period is 4
cos(kx) has period 2π/k, so k = π/2
y = cos(π/2 x)

(c)(π/3, 0) and for which the line x= π/2 is an asymptote
tan(kx) has period π/k. Your graph has half-period π/2 - π/3 = π/6, so k=3
tanx crosses the x-axis at x=0, so yours is shifted right π/3
That makes the function y = tan(3(x-π/3)) = tan(3x-π)

#2 sinx = cosx
clearly it is not true for x=0, so it is not an identity
why don't you take a stab at the other parts.
consider x^2 = 2^2...