5.3 Addition and Subtraction Identities
For angles A and B it can be proven geometrically that
sin ( A + B ) = sin A cos B + cos A sin B
and
sin ( A - B ) = sin A cos B - cos A sin B
It can also be shown that
cos ( A + B ) = cos A cos B - sin A sin B
and
cos ( A - B ) = cos A cos B + sin A sin B.
Using these results it can be shown that
tan ( A + B ) = ( tan A + tan B ) / ( 1 - tan A tan B )
and
tan ( A - B ) = ( tan A - tan B ) / ( 1 + tan A tan B ).
These are referred to as the addition and subtraction
identities.
Exercise 5.3.1
Use the exact values of the trigonometric functions of 30o and 45o and the
addition and subtraction formulas to find exact values of the
trigonometric functions of 75o and 15o.
Solution
Exercise 5.3.2
Prove the cofunction identity cos ( 90o
- A ) = sin A by using the difference identity for cosine.
Solution
Exercise 5.3.3
Find the exact value of sin 17o cos 13o + cos 17o sin
13o without using a calculator.
Solution
Exercise 5.3.4
Find an identity for tan ( x + π / 4 ).
Solution
A sum of multiples of two functions is called a linear combination
of the two functions. Thus, if f ( x ) = a sin (
x ) + b cos ( x ), then f ( x ) is a
linear combination of sin ( x ) and cos ( x ). We are going
to give an example of the general principle that every linear combination
of sin ( x ) and
cos ( x ) is a sinusoidal function.
Example: Let f ( x ) = 3 sin ( x ) + 4 cos (
x ). Show that f is a sinusoidal function. Find the
amplitude, period and phase shift.
The first step is to find the square root of the sum of the squares of the
coefficients. In this case, that would be 5. Then factor 5 out of the
expression to get
f ( x ) = 5 [ ( 3 / 5 ) sin ( x ) + ( 4 / 5 )cos (
x ) ]
Next, regard the coefficient of sin ( x ) as the cosine of some
angle P and the coefficient of cos ( x ) as the sine of the same
angle P. We can make this assumption since the sum of the squares of
the coefficients equals 1.
This gives
f ( x ) = 5 [ cos ( P ) sin ( x ) + sin ( P ) cos (
x ) ]
Using the sum formula for sine, this can be re-written
f ( x ) = 5 sin ( x + P ).
The amplitude is 5, the period is 2 ¼ and the phase shift is P.
What is the value of P ? Since both the sine and the cosine of P are
positive, it is an angle in quadrant I. So we can calculate P as arccos (
3 / 5 ) or as arcsin ( 4 / 5 ). About 0.9273 radians or 53.1o.
Exercise 5.3.5
Find the amplitude, period and phase shift of
f ( x ) = - 2 sin ( x ) + 3 cos ( x )
Solution
Return