Fundamental Trigonometric Identities Worksheets - Free Printable

It looks like you've uploaded an image of a worksheet titled "Quotient Identities" with several trigonometric problems involving the quotient identities, which are:

$$
\tan \theta = \frac{\sin \theta}{\cos \theta}, \quad \cot \theta = \frac{\cos \theta}{\sin \theta}
$$

The problems ask you to find either $\tan x$ or $\cot x$ given values of $\sin x$ and $\cos x$. Let's solve each problem step by step.

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Problem 1:

Given:
$\sin x = \frac{4}{5}$, $\cos x = \frac{3}{5}$
Find: $\tan x$

Using the identity:
$$
\tan x = \frac{\sin x}{\cos x} = \frac{4/5}{3/5} = \frac{4}{3}
$$

✔ Answer: $\boxed{\frac{4}{3}}$

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Problem 2:

Given:
$\sin x = \frac{12}{13}$, $\cos x = \frac{5}{13}$
Find: $\tan x$

$$
\tan x = \frac{\sin x}{\cos x} = \frac{12/13}{5/13} = \frac{12}{5}
$$

✔ Answer: $\boxed{\frac{12}{5}}$

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Problem 3:

Given:
$\sin x = \frac{12}{13}$, $\cos x = \frac{5}{13}$
Find: $\cot x$

$$
\cot x = \frac{\cos x}{\sin x} = \frac{5/13}{12/13} = \frac{5}{12}
$$

✔ Answer: $\boxed{\frac{5}{12}}$

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Problem 4:

Given:
$\sin x = \frac{20}{29}$, $\cos x = \frac{21}{29}$
Find: $\tan x$

$$
\tan x = \frac{20/29}{21/29} = \frac{20}{21}
$$

✔ Answer: $\boxed{\frac{20}{21}}$

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Problem 5:

Given:
$\sin x = \frac{7}{25}$, $\cos x = \frac{24}{25}$
Find: $\cot x$

$$
\cot x = \frac{\cos x}{\sin x} = \frac{24/25}{7/25} = \frac{24}{7}
$$

✔ Answer: $\boxed{\frac{24}{7}}$

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Problem 6:

Given:
$\sin x = \frac{12}{13}$, $\cos x = \frac{5}{13}$
Find: $\cot x$

Wait — this is the same as Problem 3! So:

$$
\cot x = \frac{5}{12}
$$

✔ Answer: $\boxed{\frac{5}{12}}$

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✔ Final Answers:

1. $\tan x = \boxed{\frac{4}{3}}$
2. $\tan x = \boxed{\frac{12}{5}}$
3. $\cot x = \boxed{\frac{5}{12}}$
4. $\tan x = \boxed{\frac{20}{21}}$
5. $\cot x = \boxed{\frac{24}{7}}$
6. $\cot x = \boxed{\frac{5}{12}}$

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🔍 Explanation:

- We used the quotient identities:
- $\tan x = \frac{\sin x}{\cos x}$
- $\cot x = \frac{\cos x}{\sin x}$
- Since both $\sin x$ and $\cos x$ are given as fractions with the same denominator, the denominators cancel out when dividing.
- All values are positive, so we assume $x$ is in the first quadrant, where all trig functions are positive.

Let me know if you'd like help visualizing these on the unit circle or verifying the Pythagorean identity ($\sin^2 x + \cos^2 x = 1$) for any of them!