Advanced Trigonometry

Question 1

What do sin(x + y) and sin(x - y) equal?

Answer 1

sin(x + y) = sinx cosy + cosx siny

sin(x - y) = sinx cosy - cosx siny

Question 2

What do cos(x + y) and cos(x - y) equal?

Answer 2

cos(x + y) = cos(x)cos(y) - sin(x)sin(y)

cos(x - y) = cos(x)cos(y) + sin(x)sin(y)

Question 3

What do tan(x + y) and tan(x - y) equal?

Answer 3

tan(x) + tan(y)
tan(x + y) = ——————–
1 - tan(x)tan(y)

                 tan(x) - tan(y) tan(x - y) = --------------------
                1 + tan(x)tan(y)

Question 4

What does cos(2x) equal?

Answer 4

cos^2(x) - sin^2(x)

Question 5

What does sin(2x) equal?

Answer 5

2sin(x)cos(x)

Question 6

What does tan(2x) equal?

Answer 6

1 - tan^2(x)

Question 7

How do you solve a trigonometry problem using the t results?

Answer 7

Substitute the t equations into equation.
Find t.
Find @ using:
          tan(@/2) = t
Or:
          @ = 2tan^-1(t)

Question 8

What is the t equation for tan?

Answer 8

2t
tan@ = ———-
1 - t^2

Question 9

What is the t equation for sin?

Answer 9

2t
sin@ = ———
1 + t^2

Question 10

What is the t equation for cos?

Answer 10

1 - t^2
cos@ = ———
1 + t^2

Question 11

How do you solve a problem using the auxiliary angle method?

Answer 11

Construct a triangle using the coefficients of cos and sin. The adjacent angle should be the coefficient of cos, and the opposite the coefficient of sin.
The angle in the new triangle is called ‘a’.
Find the size of a.
Multiply the whole equation by the hypotenuse and change the coefficients to cos(a) and sin(a).
Replace the brackets with cos(theta +/- a) (depending on the sign).
Solve for theta by removing a.
Find which quadrants the answer will be in.

Question 12

What is the shortcut for the auxiliary angle method?

Answer 12

asin(theta) + bcos(theta) = rsin(theta + a)

Where r = the hypotenuse of the new triangle
And a = the opposite angle that you usually mark