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Trigonometric Double-Angle and Half-Angle Formulas

Written by tutor Michael B.

Objective

In this section, you will learn formulas that establish a relationship between the basic trigonometric values (sin, cos, tan) for a particular angle and the trigonometric values for an angle that is either double- or half- of the first angle. These relationships can be very useful in proofs and also in problem solving because they can often be used to simplify an equation.

Double-Angle Formulas

Given the trigonometric values of an angle α, we would like to be able to determine the trigonometric values for another angle 2α:

Double Angle Diagram

This can be easily accomplished by realizing that 2α = α + α, and utilizing the trigonometric summation formulas. Recall the three summation formulas:

Summation Formulas

From these, we can derive the double-angle formulas for sin(2α), cos(2α), and tan(2α):

sin2a Derivation

cos2a Derivation

tan2a Derivation

In addition, the cos(2α) formula has two alternate but common forms. By utilizing the identity sin2(α) + cos2(α) = 1, we can also derive the two formulas:

cos2a Additional Forms

It is also important to note that the following relationships are NOT true:

Bad Formulas

Half-Angle Formulas

Just as with the double-angle formulas, when given the trigonometric values of an angle α, we would like to be able to determine the trigonometric values for another angle α/2:

Half Angle Diagram

By solving for sin and cos from the alternate forms of cos(2α), and then substituting α = α/2, we obtain:

sin cos Half Alpha

There is one important thing to note about these two equations. Normally, when one sees the "±" symbol in math equations, it typically means to use both the positive and the negative answer. For example, in the quadratic equation, there are two answers - one for the positive version and one for the negative version of the radical. However, in this case, only one answer (either posititive or negative) should be selected. The choice is not arbitrary – the student must use information available from the given problem to determine which answer is correct. This is typically done by determining which quadrant the angle α/2 is located in, as the sign of each trigonometric function is strictly determined by the quadrant of the angle (ASTC).

The tangent half-angle formula also has three versions that may be useful in different scenarios:

tangent half alpha formula

Examples

Example #1

Given an angle for which sin(α) = -3/5 in Quadrant III, determine the values for sin(2α), cos(2α), tan(2α), sin(α/2), cos(α/2), and tan(α/2).

Solution:

Before we begin solving for the 3 double-angle values and 3 half-angle values, let us first find cos(α) and tan(α) since they will be helpful in our calculations. By using the Pythagorean relationship and the fact that cosine is negative and tangent is positive in Quadrant III, we can determine that cos(α) = -4/5 and tan(α) = 3/4. In addition, since we know that α is in Quadrant III, we can write 180° < α=""><>, and dividing all terms of this inequality by 2, we can further write 90° < α/2=""><>. This shows that α/2 is located in Quadrant II.

Now we can easily calculate:

Find sin(2α):
sin2a

Find cos(2α):
cos2a or or


Find tan(2α):
tan2a

Find sin(α/2):
sin halfa

Find cos(α/2):
cos halfa

Find tan(α/2):
tan halfa
or

or


Example #2

Use identities to simplify and write an exact expression for each of the following using a single trigonometric function:

(a) (b) (c)
Solution:
(a)


(b)


(c)


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