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Example Questions
Example Question #1 : Identities Of Doubled Angles
Simplify the function below:
We need to use the following formulas:
a)
b)
c)
We can simplify as follows:
Example Question #1 : Identities Of Doubled Angles
Given , what is
in terms of
?
To solve this problem, we need to use the formula:
Substituting , we get
Example Question #1 : Identities Of Doubled Angles
Using trigonometric identities, determine whether the following is valid:
Only in the range of:
True
Only in the range of:
False
Uncertain
True
In order to prove this trigonometric equation we can work with either the left or right side of the equation and attempt to make them equal. We will choose to work with the left side of the equation. First we separate the fractional term:
We separated the fractional term because we notice we have a double angle. Recalling our trigonometric identities, the fractional term is the inverse of the power reducing formula for sine.
Now separating out the sine terms:
Now recalling the basic identities:
Using the trigonometric identities we have proven that the equation is true.
Example Question #1 : Identities Of Doubled Angles
Using trigonometric identities determine whether the following is true:
Only in the range of:
Uncertain
False
Only in the range of:
True
True
We choose which side to work with in the given equation. Selecting the right hand side since it contains a double angle we attempt to use the double angle formula to determine the equivalence:
Next we reduce and split the fraction as follows:
Recalling the basic identities:
This proves the equivalence.
Example Question #1 : Identities Of Doubled Angles
Using a double angle formula, find the value of
.
The formula for a doubled angle with sine is
Plug in our given value and solve.
Combine our terms.
Example Question #1 : Identities Of Doubled Angles
Find the value of if
, and if the value of
is less than zero.
Write the Pythagorean Identity.
Substitute the value of and solve for
.
Since the must be less than zero, choose the negative sign.
Write the double-angle identity of .
Substitute the known values.
Example Question #7 : Identities Of Doubled Angles
Suppose is an angle in the third quadrant, such that:
What is the value of ?
We can exploit the following trigonometric identity:
Then we can do:
With this value we can conveniently find our solution to be:
Example Question #8 : Identities Of Doubled Angles
What is the period of ?
The key here is to double-angle identity for to simplify the function.
In this case, , which means...
From there, we can use the fact that the period of or
is
. Consequently,
Example Question #1 : Identities Of Doubled Angles
Expand the following expression using double-angle identities.
Since
and ,
then .
Here we have to use the double-angle identities for both sine and cosine,
and
.
Using these identities:
Using the distributive property:
Example Question #1 : Identities Of Doubled Angles
Simplify the function below:
We need to use the following formulas:
a)
b)
c)
d)
e)
We can simplify as follows:
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