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Example Questions
Example Question #1 :Pythagorean Identities
Simplify.
To simplify, recognize thatis a reworking on,这意味着.
Plug that into our given equation:
Remember that, so.
Example Question #1 :Pythagorean Identities
Simplify.
Recognize thatis a reworking on,这意味着.
Plug that in to our given equation:
Notice that one of the's cancel out.
.
Example Question #2 :Trigonometry
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 :Pythagorean Identities
Simplify the equation using identities:
1
There are a couple valid strategies for solving this problem. The simplest is to first factor outfrom both sides. This leaves us with:
Next, substitute with the known identityto get:
From here, we can eliminate the quadratic by converting:
giving us
Thus,
Example Question #2 :Pythagorean Identities
Simplify the expression:
The equation cannot be further simplified.
The expressionrepresents adifference 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 #6 :Trigonometric Identities
If theta is in the second quadrant, and, what is?
Write the Pythagorean Identity.
Substitute the value ofand solve for.
Since the cosine is in the second quadrant, the correct answer is:
Example Question #7 :Trigonometric Identities
For which values ofis 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 byyields:
.
Dividing both sides byyields:
.
This is precisely the definition of the tangent function; since the domain ofconsists of all real numbers, the values ofwhich 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 byallows us to simplify the right-hand side.
Example Question #9 :Trigonometric Identities
What isequal to?
Step 1: Recall the trigonometric identity that has sine and cosine in it...
The sum is equal to 1.
Example Question #1 :Pythagorean Identities
Given, what is?
Using the Pythagorean Identity
,
one can solve forby plugging infor.
Solving for, you get it equal to.
Taking the square root of both sides will get the correct answer of
.
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