Hi there. In this math post I cover the double angle formula for the sine trigonometric function. I cover the base formula, a rearranged formula and cover some examples. It is assumed that the reader is familiar with basic trigonometric rations, special right angle triangles and the CAST rule.
Math pictures rendered in LaTeX along with using Quicklatex.com.
Double Angle Formula For Sine
The double angle for the sine function is useful when double the angle is not a common angle like 30°, 60°, 90°, 120° and so on. These common angles are nice to work with for sine, cosine, tangent, etc if a calculator is not allowed.
For the sine function, the double angle formula takes double of an angle theta (θ) as an input for sine. This sine of the double angle is equal to two times sine of the angle theta (θ) times cosine of the angle theta (θ).
Rearranged Version Of Sine Double Angle Formula
Lately I have been using a rearranged version of the sine double angle formula for trigonometric proofs and algebra for some grade 12 students at my work. It is not easy to even think about using this rearranged formula or think that this could be helpful. Here it is.
If you have just sine of an angle multiplied by cosine of the same angle this is equal to sine of double the angle then divide by 2.
Double Angle Formula For Sine From Sum Of Angle Formula
The quantity 2x can be viewed as x + x. For the double angle formula for sine the angle 2θ can be viewed as θ + θ. The double angle formula is derived from the sum of angle formula.
Examples
Example One
Given that the angle theta θ is equal to 30 degrees, what is the sine of double of 30 degrees?
Example Two
If sin(A) is equal to 5/13 and cos(A) is -12/13 what is the value of sin(2A)?
Start with the double angle formula for sine.
In this example you do not need to find the value of the angle A. You just want the sine of twice the angle A.
Example Three
Evaluate the product of sine of 15 degrees and the cosine of 15 degrees without a calculator.
This particular example is a weird one. You have to use a rearranged version of the double angle formula for sine. The regular double angle formula for sine is as follows.
You have the sine of 15 degrees and the cosine of 15 degrees. The multiplier of 2 from the right side can go to the left side in the form of a half (divide by 2 both sides). Another way to write the double angle formula for sine is with A = 15 degrees is:
The sine of 30 degrees is known as it is one half. Thirty degrees is a common angle for trigonometry.
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