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Example Questions
Example Question #1 : Trigonometric Identities
Simplify .
To simplify, recognize that is a reworking on
, meaning that
.
Plug that into our given equation:
Remember that , so
.
Example Question #1 : Pythagorean Identities
Simplify .
Recognize that is a reworking on
, meaning that
.
Plug that in to our given equation:
Notice that one of the 's cancel out.
.
Example Question #1 : Trigonometric Identities
0
1
-1
1
Recall the Pythagorean Identity:
We can rearrange the terms:
This is exactly what our original equation looks like, so the answer is 1.
Example Question #1 : Trigonometric Identities
Simplify the equation using identities:
1
There are a couple valid strategies for solving this problem. The simplest is to first factor out from both sides. This leaves us with:
Next, substitute with the known identity to get:
From here, we can eliminate the quadratic by converting:
giving us
Thus,
Example Question #1 : Pythagorean Identities
Simplify the expression:
The equation cannot be further simplified.
The expression represents a difference of squares. In this case, the product is
(remember that 1 is also a perfect square).
One Pythagoran identity for trigonometric functions is:
Thus, we can say that the most simplified version of the expression is .
Example Question #1 : Trigonometric Identities
If theta is in the second quadrant, and , what is
?
Write the Pythagorean Identity.
Substitute the value of and solve for
.
Since the cosine is in the second quadrant, the correct answer is:
Example Question #7 : Trigonometric Identities
For which values of is the following equation true?
According to the Pythagorean identity
,
the right hand side of this equation can be rewritten as . This yields the equation
.
Dividing both sides by yields:
.
Dividing both sides by yields:
.
This is precisely the definition of the tangent function; since the domain of consists of all real numbers, the values of
which satisfy the original equation also consist of all real numbers. Hence, the correct answer is
.
Example Question #8 : Trigonometric Identities
By the Pythagorean identity, the first two terms simplify to 1:
.
Dividing the Pythagorean identity by allows us to simplify the right-hand side.
Example Question #1 : Trigonometric Identities
What is equal to?
Step 1: Recall the trigonometric identity that has sine and cosine in it...
The sum is equal to 1.
Example Question #1 : Trigonometric Identities
Given , what is
?
Using the Pythagorean Identity
,
one can solve for by plugging in
for
.
Solving for , you get it equal to
.
Taking the square root of both sides will get the correct answer of
.
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