Half-angle identities – Formulas, proof and examples

Half-angle identities are trigonometric identities used to simplify trigonometric expressions and calculate the sine, cosine, or tangent of half-angles when we know the values of a given angle. These identities are obtained by using the double angle identities and performing a substitution.

Here, we will learn to derive the half-angle identities and apply them to solve some practice exercises.

TRIGONOMETRY
half-angle identities in trigonometry

Relevant for

Learning about half-angle identities with examples.

See examples

TRIGONOMETRY
half-angle identities in trigonometry

Relevant for

Learning about half-angle identities with examples.

See examples

What are the half-angle identities?

Half-angle identities are trigonometric identities that are used to calculate or simplify half-angle expressions, such as \sin(\frac{\theta}{2}). These identities can also be used to transform trigonometric expressions with exponents to one without exponents.

The half-angle identity of the sine is:

\sin(\frac{\theta}{2})=\pm \sqrt{\frac{1-\cos(\theta)}{2}}

The half-angle identity of the cosine is:

\cos(\frac{\theta}{2})=\pm \sqrt{\frac{1+\cos(\theta)}{2}}

The half-angle identity of the tangent is:

\tan(\frac{\theta}{2})=\frac{\sin(\theta)}{1+\cos(\theta)}

=\frac{1-\cos(\theta)}{\sin(\theta)}

Proof of the half-angle identities

The mean angle identities can be derived using the double angle identities.

To derive the formula for the identity of half-angle of sines, we start with the double angle identity of cosines:

\cos(2\theta)=1-2{{\sin}^2}(\theta)

If we use the relation \theta=\frac{\alpha}{2}, we have 2\theta=\alpha. Substituting these expressions in the identity above, we have:

\cos(\alpha)=1-2{{\sin}^2}(\frac{\alpha}{2})

Now, we solve this expression for \sin(\frac{\alpha}{2}):

\cos(\alpha)=1-2{{\sin}^2}(\frac{\alpha}{2})

2{{\sin}^2}(\frac{\alpha}{2})=1-\cos(\alpha)

{{\sin}^2}(\frac{\alpha}{2})=\frac{1-\cos(\alpha)}{2}

\sin(\frac{\alpha}{2})=\pm\sqrt{\frac{1-\cos(\alpha)}{2}}

The sign of \sin(\frac{\alpha}{2}) depends on the quadrant in which \frac{\alpha}{2} is located. If \frac{\alpha}{2} is in the first or second quadrant, the formula uses the positive sign, and if \frac{\alpha}{2} is in the third or fourth quadrant, the formula uses the negative sign.

We use a similar process to find the half-cosine angle identity. Therefore, we start with the double-angle identity of the cosine in the following form:

\cos(2\theta)=2{{\cos}^2}(\theta)-1

After making the substitutions, we get:

\cos(\alpha)=2{{\cos}^2}(\frac{\alpha}{2})-1

Now, we solve for \cos(\frac{\alpha}{2}):

\cos(\alpha)=2{{\cos}^2}(\frac{\alpha}{2})-1

2{{\cos}^2}(\frac{\alpha}{2})=1+\cos(\alpha)

{{\cos}^2}(\frac{\alpha}{2})=\frac{1+\cos(\alpha)}{2}

\cos(\frac{\alpha}{2})=\pm\sqrt{\frac{1+\cos(\alpha)}{2}}

In this case, if \frac{\alpha}{2} is in the first or fourth quadrant, the formula uses the positive sign and if \frac{\alpha}{2} is in the second or third quadrant, the formula uses the negative sign.


Half-angle identities – Example with answers

The following examples are solved using what you have learned about half-angle identities. Study and analyze these examples to understand the process used.

EXAMPLE 1

Use the half-angle identity of the sine to find the sine value of 15°.

We use the formula for the half-angle identity of the sine with the given value. Therefore, we have:

\sin(\frac{\theta}{2})=\pm\sqrt{\frac{1-\cos(\theta)}{2}}

\sin(15^{\circ})=\pm\sqrt{\frac{1-\cos(30^{\circ} )}{2}}

=\pm\sqrt{\frac{1-0.866}{2}}

=0.259

We use the positive value since 15° is in the first quadrant.

EXAMPLE 2

Determine the value of the cosine of 165° using the half-angle identity of the cosine.

To use the half-angle identity of cosine, we use the angle \frac{\theta}{2} = 165°. This means that we have \theta = 330°. Therefore, we use the formula with these values:

\cos(\theta)=\pm\sqrt{\frac{1+\cos(\theta)}{2}}

\cos(165^{\circ})=\pm\sqrt{\frac{1+\cos(330^{\circ})}{2}}

=\pm\sqrt{\frac{1+0.866}{2}}

=-0.966

We chose the negative value since the angle 165° is in the second quadrant.

EXAMPLE 3

Check that the identity 2{{\sin}^2}(\frac{x}{2})+\cos(x)= 1.

We can use the identity of the half-angle of sine to substitute and simplify the expression. By doing this, we have:

2{{\sin}^2}(\frac{x}{2})+\cos(x)

=2{{(\sqrt{\frac{1-\cos(x)}{2}})}^2}+\cos(x)

=2(\frac{1-\cos(x)}{2})+\cos(x)

=1-\cos(x)+\cos(x)

=1

After simplifying, we see that the left side of the identity is equal to the right side, so the identity is true.


Half-angle identities – Practice problems

Solve the following practice problems using what you have learned about the half-angle identities of sine, cosine, and tangent. Select an answer and check it to see if you got the correct answer.

Find the value of \sin(75^{\circ}) using half-angle identities

Choose an answer






If we have \cos(A)=\frac{12}{13}, where angle A is between 0° and 90°, what is the value of \sin(\frac{A}{2})?

Choose an answer







See also

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