Pythagorean Identities

Answer: $\sin^2\theta + \cos^2\theta = 1$

The fundamental Pythagorean identity is $\sin^2\theta + \cos^2\theta = 1$. This is derived from the unit circle definition where $x = \cos\theta$ and $y = \sin\theta$.

Answer: Second Pythagorean identity

$1 + \tan^2\theta = \sec^2\theta$ is the second Pythagorean identity. It is derived by dividing $\sin^2\theta + \cos^2\theta = 1$ by $\cos^2\theta$.

Answer: 0.64

Using $\sin^2\theta + \cos^2\theta = 1$: $0.36 + \cos^2\theta = 1$, so $\cos^2\theta = 1 - 0.36 = 0.64$.

Answer: 1

From $1 + \tan^2\theta = \sec^2\theta$, we can rearrange to get $\sec^2\theta - \tan^2\theta = 1$.

Answer: $\frac{12}{13}$

$\left(\frac{5}{13}\right)^2 + \cos^2\theta = 1$ gives $\frac{25}{169} + \cos^2\theta = 1$, so $\cos^2\theta = \frac{144}{169}$ and $\cos\theta = \frac{12}{13}$ (positive in Q1).

Answer: -3/5

$\sin^2\theta + \left(\frac{4}{5}\right)^2 = 1$ gives $\sin^2\theta = 1 - \frac{16}{25} = \frac{9}{25}$. Since $\theta$ is in Q4, $\sin\theta = -\frac{3}{5}$.

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 $\frac{\sin^2\theta}{\sin^2\theta}$ become? 1
4. What is $\frac{\cos^2\theta}{\sin^2\theta}$ in terms of cotangent? $\cot^2\theta$

Answer: $1$

From $1 + \cot^2\theta = \csc^2\theta$, we get $\csc^2\theta - \cot^2\theta = 1$.

Answer: $-\frac{5}{4}$

1. Which Pythagorean identity relates tangent and secant? $1 + \tan^2\theta = \sec^2\theta$
2. What is $\tan^2\theta$ if $\tan\theta = \frac{3}{4}$? 9/16
3. What is $\sec^2\theta = 1 + \frac{9}{16}$? 25/16
4. In Quadrant III, is secant positive or negative? negative
5. So what is $\sec\theta$? Enter it as a fraction. -5/4

Answer: $1 - \cos\theta$

1. Which expression can replace $\sin^2\theta$ in the numerator? $1 - \cos^2\theta$
2. Factor $1 - \cos^2\theta$ as a difference of squares: $(1 - \cos\theta)(1 + \cos\theta)$
3. Cancel the common factor $1 + \cos\theta$. What is left? $1 - \cos\theta$
4. Check with $\theta = 60°$, where $\cos(60°) = 0.5$: what is $1 - \cos(60°)$? 0.5

Answer: $\tan^2\theta - \sin^2\theta = \tan^2\theta \cdot \sin^2\theta$

1. Express $\tan^2\theta$ in terms of sine and cosine: $\frac{\sin^2\theta}{\cos^2\theta}$
2. Substitute into the left side: $\frac{\sin^2\theta}{\cos^2\theta} - \sin^2\theta$. Which common factor can you take out? $\sin^2\theta$
3. After factoring: $\sin^2\theta\left(\frac{1}{\cos^2\theta} - 1\right)$. Write the bracket as one fraction: $\frac{1 - \cos^2\theta}{\cos^2\theta}$
4. Use $1 - \cos^2\theta = \sin^2\theta$. What does the left side become? $\tan^2\theta \cdot \sin^2\theta$

Answer: $\frac{12}{25}$

$(\sin\theta + \cos\theta)^2 = \sin^2\theta + 2\sin\theta\cos\theta + \cos^2\theta = 1 + 2\sin\theta\cos\theta = \frac{49}{25}$. So $2\sin\theta\cos\theta = \frac{49}{25} - 1 = \frac{24}{25}$, and $\sin\theta\cos\theta = \frac{12}{25}$.

Answer: $2\sec^2\theta$

1. What is the common denominator? $(1 - \sin\theta)(1 + \sin\theta)$
2. Multiply out the denominator (difference of squares): $1 - \sin^2\theta$
3. What is $1 - \sin^2\theta$ equal to? $\cos^2\theta$
4. What is the numerator after adding the fractions? 2
5. So the sum is $\frac{2}{\cos^2\theta}$. Which expression is that? $2\sec^2\theta$

Answer: $1$

By the fundamental identity, $\sin^2\theta + \cos^2\theta = 1$ for any angle $\theta$. Even though $\sin(45°) = \cos(45°) = \frac{\sqrt{2}}{2}$, the sum of their squares is always 1.

Answer: 13/12

$1 + \left(\frac{5}{12}\right)^2 = \csc^2\theta$ gives $1 + \frac{25}{144} = \frac{169}{144}$. So $\csc\theta = \frac{13}{12}$ (positive in Q1).