In other Concepts you've dealt with double angle formulas. This was useful for finding the value of an angle that was double your well known value. Now consider the idea of a "triple angle formula". If someone gave you a problem like this:
Could you compute its value?
Keep reading, and at the end of this Concept you'll know how to simplify equations such as this using the triple angle formula.
Deriving a Triple Angle Formula
Double angle formulas are great for computing the value of a trig function in certain cases. However, sometimes different multiples than two times and angle are desired. For example, it might be desirable to have three times the value of an angle to use as the argument of a trig function.
By combining the sum formula and the double angle formula, formulas for triple angles and more can be found.
Here, we take an equation which takes a linear combination of sine and cosine and converts it into a simpler cosine function.
, where , and .
You can also use the TI-83 to solve trigonometric equations. It is sometimes easier than solving the equation algebraically. Just be careful with the directions and make sure your final answer is in the form that is called for. You calculator cannot put radians in terms of .
Find the formula for
Solution: Use both the double angle formula and the sum formula.
Transform into the form
Solution: and , so . Therefore and which makes the reference angle is or -0.927 radians. since cosine is positive and sine is negative, the angle must be a fourth quadrant angle. must therefore be or 5.36 radians.The final answer is .
Solve such that using a graphing calculator.
Solution: In , graph and .
Next, use CALC to find the intersection points of the graphs.
Linear Combination: A linear combination is a set of terms that are added or subtracted from each other with a multiplicative constant in front of each term.
Triple Angle Identity: A triple angle identity relates the a trigonometric function of three times an argument to a set of trigonometric functions, each containing the original argument.
1. Transform to the form
2. Transform to the form
3. Derive a formula for .
1. If , then and . By the Pythagorean Theorem, and . So, because is negative, is in Quadrant IV. Therefore, . Our final answer is .
2. If , then and . By the Pythagorean Theorem, . Because and are both negative, is in Quadrant III, which means rad. Our final answer is .
Concept Problem Solution
Using the triple angle formula we learned in this Concept for the sine function, we can break the angle down into three times a well known angle:
we can solve this problem.
Transform each expression to the form .
Derive a formula for each expression.
Find all solutions to each equation in the interval .