Trigonometry - Double and Half Formulas

Trigonometry - Double and Half Formulas - How it Works - Video

Unit Circle

Unit Circle:

The unit circle is vital to answer any exact answer. Do you have to memorize every bit? That is up to. It would be faster if you had it encoded in your brain, but you can memorized just the first quadrant and use symmetry to fill out the unit circle.

Double Angle Formulas

Double Angle Formulas:

Here we have the formulas.

Cos(2u) has three different options. We can use the Pythagorean Identity, sin2u + cos2u = 1, to substitute to find the other two.

Half-Angle Identities and Half-Angle Formulas

Half-Angle Identities and Half-Angle Formulas:

Here we have the formulas.

Tan(u/2) has two different options. We can multiply by the conjugate of 1 - cos(u), which is 1 + cos(u), to the numerator and denominator. After using the Pythagorean Identity, sin2u + cos2u = 1, we can simplify to get the second.

Remember there tan = sin / cos. So there is a third option, where we have square roots, but it looks cleaner without.

Example 1

Example 1:

Our first step is to triangle in Q III, because of the given, 180° < θ < 270°.

Now we have to decided where the legs, 3 and 4, go. We know that they are legs because cotangent = adjacent / opposite. We know tangent flips that so tan(θ) = 3/4.

Theta, θ, needs to be write where it is, because we were told to go in the counterclockwise direction from the given, 180° < θ < 270°. This means that 3 is across the angle and 4 touches the angle.

Remember those are distances, which are positive, but when we write down what sin(θ), cos(θ), and tan(θ), we need to use the point (-4, -3). Using the Pythagorean Theorem, we can find the hypotenuse to be 5.

So we have sin(θ) = -3/5, cos(θ) = -4/5, and tan(θ) = 3/4.

Example 1 continued:

Now we know θ or u in this case. We can substitute each one in the correct formula.

tan(2u) = [2 * tan(u)] / [1 - tan2 (u)] sin(2u) = 2 sin(u) * cos(u) cos(2u) = cos2 (u) - sin2 (u) or any of the three options

tan(2u) = [2 * tan(u)] / [1 - tan2 (u)]

= (2 * [3/4]) / (1 - [3/4]2)

= [3/2] / (1 - [9/16])

= [3/2] / [7/16]

= [3/2] * [16/7]

=24/7


Substituted for u.

Multiplied the numerator and squared.

Subtracted the denominator.

Flipped the fraction.

Simplified.

sin(2u) = 2 sin(u) * cos(u)

= 2 * [-3/5] * [-4/5]

=2 * [12/25]

= 24/25


Substituted for u.

Multiplied the fractions.

Multiplied the two numbers.

cos(2u) = cos2 (u) - sin2 (u)

= [-4/5]2 - [-3/5]2

= [16/25] - [9/25]

= 7/25


Substituted for u.

Squared the fractions.

Subtracted the fractions.

Example 1 continued:

Now we know tan(2u) = 24/7, sin(2u) = 24/25, and cos(2u) = 7/25.

Example 2

Example 2:

We are given the original angle. This angle is considered the half angle so we need to multiply the given angle by 2 so that we can use the formula since the formula requires the full angle.

For example 2A, we have cos(157°30'), so u/2 = 157°30'. When we multiply the given angle by 2 to the whole angle, we get 315°. 315° has an exact value on the unit circle at the point, (sqrt(2)/2, -sqrt(2)/2).

Line 1 - Substituted 315° for u and chose the negative formula because 157°30' is in Q II and x is negative.

Line 2 - Substituted sqrt(2)/2 for cos(315°).

Line 3 - Changed 1 to 2/2 to have a common denominator.

Line 4 - Combined fractions.

Line 5 - Flipped the fraction.

Line 6 - Multiplied fractions.

Line 7 - Split the fractions.

Line 8 - Squared rooted the denominator.

So answer is - sqrt[2 + sqrt(2)] / 2.

Example 2:

We are given the original angle. This angle is considered the half angle so we need to multiply the given angle by 2 so that we can use the formula since the formula requires the full angle.

For example 2b, we have tan(π/8), so u/2 = π/8. When we multiply the given angle by 2 to the whole angle, we get π/4. 315° has an exact value on the unit circle at the point, (sqrt(2)/2, sqrt(2)/2).

Line 1 - Substituted π/4 for u.

Line 2 - Substituted sqrt(2)/2 for both sine and cosine.

Line 3 - Rewrote line 2 and added brackets.

Line 4 - Multiplied the numerator and denominator by 2 to get rid of the denominators of the top and bottom fraction.

Line 5 - Rewrote line 4 and added brackets.

Line 6 - Multiplied the numerator and denominator by sqrt(2) to get rid of the square root in the denominator.

Line 7 - Factored 2 out and simplified.

So answer is sqrt(2) - 1.

Live Worksheet - Double-Angle Formulas

Live Worksheet - Half-Angle Formulas

Teacher - Edpuzzle Link