cos (A - B) = (cos A)(cos B) = (sin A)(sin B)
The accompanying figure gives a graphical representation of the cosine identity.
This identity is useful when you are asked to find the cosine of a non-30°-45°-60°-90° angle, such as 10°.
Example:
1. Problem: Find the cos 15°.
Solution: Write 15° in terms of angles with known trig.
ratio values.
cos (45° - 30°)
Use the cosine identity to rewrite the expression.
(cos 45°)cos 30° + (sin 45°)sin 30°
Using the values you know for the trig. ratios of special angles,
rewrite the expression.
SQRT(2) SQRT(3) SQRT(2) 1
------- * ------- + ------- * -
2 2 2 2
Perform the indicated multiplications.
SQRT(6) SQRT(2)
------- + -------
4 4
SQRT(6) + SQRT(2)
-----------------
4
There is also a cosine identity for a sum of angles. It is shown
below.
cos (A + B) = (cos A)cos B - (sin A)sin B
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sin (A + B) = (sin A)(cos B) + (cos A)(sin B)
sin (A - B) = (sin A)(cos B) - (cos A)(sin B)
tan (A + B) = (tan A + tan B)/(1 - (tan A)(tan B))
tan (A - B) = (tan A - tan B)/(1 + (tan A)(tan B))
Example:
1. Problem: Find tan 15°.
Solution: Rewrite as a difference of angles with known trig. ratio values.
tan (45° - 30°)
Use the tangent of differences identity to rewrite the
expression.
tan 45° - tan 30°
--------------------
1 + (tan 45°)(tan 30°)
Substitute the known trig. ratio values in and perform the
indicated operations.
1 - (SQRT(3))/3
---------------
1 + (SQRT(3))/3
3 - SQRT(3)
-----------
3 + SQRT(3)
2 - SQRT(3)
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sin 2x = sin (x + x)
Use the sine of sums identity.
(sin x)(cos x) + (cos x)(sin x)
2(sin x)(cos x)
Example:
1. Problem: If the sine of theta is (3/8) and theta is in the first
quadrant, what is sin 2(theta)? (Use the
accompanying figure.)
Solution: From the diagram, we see that cos (theta) = (SQRT(55)/8.
Use the double angle identity for sine.
sin 2(theta) = 2(sin (theta))(cos (theta))
Plug in the values you know.
2 * (3/8) * (SQRT(55)/8)
Perform the indicated multiplications.
(3(SQRT(55)))/32
The other identities are listed below.
cos 2x = cos^2 x - sin^2 x
cos 2x = 1 - 2sin^2 x
cos 2x = 2cos^2 x - 1
tan 2x = (2tan x)/(1 - tan^2 x)
sin^2 x = (1 - cos 2x)/2
cox^2 x = (1 + cos 2x)/2
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Example:
1. Problem: Prove the following identity:
tan^2 x - sin^2 x = (sin^2 x)(tan^2 x)
Solution: Write each side in terms of sin x and cos x.
sin^2 x sin^2 x
------- - sin^2 x = (sin^2 x)-------
cos^2 x cos^2 x
Now, only deal with one side of the equation. Find the common
denominator and subtract.
sin^2 x - (sin^2 x)(cos^2 x)
----------------------------
cos^2 x
Factor out a sin^2 x.
(sin^2 x)(1 - cos^2 x)
----------------------
cos^2 x
Use the Pythagorean Identities
to replace (1 - cos^2 x) with sin^2 x.
(sin^2 x)(sin^2 x)
------------------
cos^2 x
sin^2 x
(sin^2 x)-------
cos^2 x
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Take the quiz on trigonometric identities. The quiz is very useful for either review or to see if you've really got the topic down.