All Trigonometry Resources
Example Question #1 : Identities Of Halved Angles
Simplify the function below:
We need to use the following formulas:
We can simplify as follows:
Example Question #45 : Trigonometric Identities
Find if and .
The double-angle identity for sine is written as
and we know that
Using , we see that , which gives us
Since we know is between and , sin is negative, so . Thus,
Finally, substituting into our double-angle identity, we get
Example Question #2 : Identities Of Halved Angles
Find the exact value of using an appropriate half-angle identity.
The half-angle identity for sine is:
If our half-angle is , then our full angle is . Thus,
The exact value of is expressed as , so we have
Simplify under the outer radical and we get
Now simplify the denominator and get
Since is in the first quadrant, we know sin is positive. So,
Example Question #3 : Identities Of Halved Angles
Which of the following best represents ?
Write the half angle identity for cosine.
Replace theta with two theta.
Example Question #4 : Identities Of Halved Angles
What is the amplitude of ?
The key here is to use the half-angle identity for to convert it and make it much easier to work with.
In this case, , so therefore...
Consequently, has an amplitude of .
Example Question #5 : Identities Of Halved Angles
If , then calculate .
Because , we can use the half-angle formula for cosines to determine .
For this problem,
Example Question #6 : Identities Of Halved Angles
What is ?
Let ; then
We'll use the half-angle formula to evaluate this expression.
Now we'll substitute for .
is in the first quadrant, so is positive. So
Example Question #7 : Identities Of Halved Angles
What is , given that and are well defined values?
Using the half angle formula for tangent,
we plug in 30 for .
We also know from the unit circle that is and is .
Plug all values into the equation, and you will get the correct answer.