Pythagorean Identities

Show your work for each problem.

1.
What is the fundamental Pythagorean identity?
1. a)
2. b)
3. c)
4. d)
Answer:
The fundamental Pythagorean identity is . This is derived from the unit circle definition where and .
2.
Which identity is ?
1. a)Quotient identity
2. b)First Pythagorean identity
3. c)Second Pythagorean identity
4. d)Third Pythagorean identity
Answer: Second Pythagorean identity
is the second Pythagorean identity. It is derived by dividing by .
3.
If , what is ?
Answer: 0.64
Using : , so .
4.
Simplify:
Answer: 1
From , we can rearrange to get .
5.
If and is in Quadrant I, what is ?
1. a)
2. b)
3. c)
4. d)
Answer:
gives , so and (positive in Q1).
6.
If and is in Quadrant IV, find . Enter as a fraction (e.g., -3/5).
Answer: -3/5
gives . Since is in Q4, .
7.
Derive from the fundamental identity.
Answer: 1 + cot^2(theta) = csc^2(theta)
1. Start with the fundamental identity. What is it? sin^2(theta) + cos^2(theta) = 1
2. To get cotangent and cosecant, divide both sides by which function squared? sin^2(theta)
3. After dividing, what does become? 1
4. What is in terms of cotangent? cot^2(theta)
8.
Simplify:
1. a)
2. b)
3. c)
4. d)
Answer:
From , we get .
9.
If and is in Quadrant III, find .
Answer: -5/4
1. Which Pythagorean identity relates tangent and secant? 1 + tan^2(theta) = sec^2(theta)
2. What is if ? 9/16
3. What is ? 25/16
4. In Quadrant III, is secant positive or negative? negative
10.
Simplify: (Hint: Use the identity to factor)
Answer: 1 - cos(theta)
11.
Verify the identity:
Answer: verified
1. Express in terms of sine and cosine: sin^2(theta)/cos^2(theta)
2. Substitute into the left side: . What common factor can you create? sin^2(theta)
3. After factoring: . Simplify the parentheses. (1 - cos^2(theta))/cos^2(theta)
4. Use to get the final form. tan^2(theta) * sin^2(theta)
12.
If , what is ?
1. a)
2. b)
3. c)
4. d)
Answer:
. So , and .
13.
Simplify:
Answer: 2sec^2(theta)
1. What is the common denominator? (1 - sin(theta))(1 + sin(theta))
2. Simplify the denominator using difference of squares: 1 - sin^2(theta)
3. What is equal to? cos^2(theta)
4. What is the numerator after adding the fractions? 2
14.
What is ?
1. a)
2. b)
3. c)
4. d)
Answer:
By the fundamental identity, for any angle . Even though , the sum of their squares is always 1.
15.
If and is in Quadrant I, find . Enter as a fraction.
Answer: 13/12
gives . So (positive in Q1).