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Example Questions
Example Question #51 :Trigonometry
Findifand.
The double-angle identity for sine is written as
and we know that
Using, we see that, which gives us
Since we knowis betweenand, sinis negative, so. Thus,
.
Finally, substituting into our double-angle identity, we get
Example Question #1 :Identities Of Halved Angles
Find the exact value ofusing 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 ofis expressed as, so we have
Simplify under the outer radical and we get
Now simplify the denominator and get
Sinceis in the first quadrant, we know sin is positive. So,
Example Question #53 :Trigonometric Identities
Which of the following best represents?
Write the half angle identity for cosine.
Replace theta with two theta.
Therefore:
Example Question #54 :Trigonometric Identities
What is the amplitude of?
The key here is to use the half-angle identity forto convert it and make it much easier to work with.
In this case,,因此……
Consequently,has an amplitude of.
Example Question #55 :Trigonometric Identities
If, then calculate.
Because, we can use the half-angle formula for cosines to determine.
In general,
for.
For this problem,
Hence,
Example Question #56 :Trigonometric Identities
What is?
Let; then
.
We'll use the half-angle formula to evaluate this expression.
Now we'll substitutefor.
is in the first quadrant, sois positive. So
.
Example Question #51 :Trigonometric Identities
What is, given thatandare well defined values?
Using the half angle formula for tangent,
,
we plug in 30 for.
We also know from the unit circle thatisandis.
Plug all values into the equation, and you will get the correct answer.
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