Associated Topics || Dr. Math Home || Search Dr. Math

### Calculating Cosine Values Graphically and Algebraically

```
Date: 05/01/98 at 14:31:10
From: Vern Lindsey
Subject: Trigonometry

I'm trying to upgrade myself and I need more math. Just looking for
help any where I can find it. I took college algebra, and I now need
trig. I am self teaching before I take the class.

I'm studying out of College Algebra and Trigonometry by Linda L. Exley
and Vincent K. Smith, Dekalb College. Chapter 9 Section 9.3, about
sine and cosine functions, lists problem #55:

Suppose cos(t) = -2/5.
Find cos(-t), sin(pi/2 - t), cos(t + 2pi), and cos(t - 2pi)

I see that cos(-t) = -2/5 by definition, since cos(-t) = cos(t).
However, I'm not seeing what I'm supposed to see on the others. I've
tried the fundamental identity cos^2(t) + sin^2(t) = 1, where cos^2(t)
is actually read cosine squared t. Anyway I'm lost as to what the
lesson wants me to see here. Can you help? Thanks.
```

```
Date: 05/01/98 at 17:08:09
From: Doctor Sam
Subject: Re: Trigonometry

Vern,

I'm not sure where you are in your study of trigonometry, so I'll
suggest two different ways to evaluate these trig functions. One
method is algebraic and uses the subtraction identities for sines and
cosines. The other method is graphical and will let you visualize the
solutions.

Graphical

The graphs of y = sin(t) and y = cos(t) are very similar. Both have
period 2pi. Physicists say that the two waves are "pi/2 radians out of
phase." A mathematician might say that you can get the graph of
y = cos(t) by shifting the graph of y = sin(t) by pi/2 radians to
the left.

In fact, if you look at the graphs of y = sin(t) and y = cos(t), you
will see the effect of shifting y = sin(t) by multiples of pi/2:

one shift:     sin(t + pi/2)   = cos(t)
two shifts:    sin(t + pi)     = - sin(t)
three shifts:  sin(t + 3pi/2)  = - cos(t)
four shifts:   sin(t + 2pi)    = cos(t)

That's all we need to compute the desired values.

1.  sin(pi/2 - t) = sin(-[t - pi/2]) = - sin(t - pi/2)
This shifts the sine graph pi/2 units to the right,
which is the same as shifting it 3pi/2 units to the
left. So

- sin(t - pi/2) = - [- cos(t)] = cos(t) = -2/5

2.  cos(t + 2pi) = cos(t) = -2/5 because 2pi is the
period of cos(t).

3.  cos(t - 2pi) = cos(t) = -2/5 for the same reason.

Algebraic

Somewhere in your book, you will find several useful trig identities
called addition formulas. They are used to change things like
sin(A + B) into expressions involving only sines and cosines of A and
B separately. Here they are:

sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
and
cos(A + B) = cos(A)cos(B) - sin(A)sin(B)

A and B can be positive or negative.

1.  sin(pi/2 - t). Use A = pi/2, and B = -t:
sin(pi/2 - t) = sin(pi/2)cos(t) + cos(pi/2)sin(t)

Now sin(pi/2) = 1, and cos(pi/2) = 0, so
sin(pi/2 - t) = cos(t) = -2/5

2.  cos(t + 2pi). Use A = t, and B = 2pi
cos(t + 2pi) = cos(t)cos(2pi) - sin(t)sin(2pi)

Since sin(2pi) = 0 and cos(2pi) = 1,
cos(t + 2pi) = cos(t) = -2/5

3.  cos(t - 2pi). Use A = t, and B = -2pi.
cos(t - 2pi) = cos(t)cos(-2pi) - sin(t)sin(-2pi)

Since cos(-2pi) = 1 and sin(-2pi) = 0,
cos(t - 2pi) = cos(t) = -2/5

I hope that helps!

-Doctor Sam, The Math Forum
Check out our web site! http://mathforum.org/dr.math/
```
Associated Topics:
High School Trigonometry

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search