Given below are two statement: Statement I: If ()=5/13, then the value of ()=5/12 Statement II: if ()=12/5, then the value of ()=5/12 In the light of the above statements, choose the correct answer form the question below: Option 1: Both Statement I and Statement II are true Option 2: Both Statement I and Statement II are false Option 3: Statement I is true but Statement II is false Option 4: Statement I is false but Statement II is true

Answer: Statement I is true but Statement II is false. Solution: Statement I: If () = 5/13 In right triangle: ^2() + ^2() = 1 (5/13)^2 + ^2() = 1 ^2() = 1 - 25/169 = 144/169 () = 12/13 Therefore () = (())/(()) = 5/13 × 13/12 = 5/12 ✓ Statement II: If () = 12/5 () = (())/(()) = 12/5 Therefore () = 5/13 (using ^2() + ^2() = 1 again) Not equal to 5/12 ✗ Therefore, Statement I is true and Statement II is false.

Geometry

Trigonometry

Easy

Given below are two statement:

Statement I: If $\sin (\theta)=\frac{5}{13}$ , then the value of $\tan (\theta)=\frac{5}{12}$

Statement II: if $\cot (\theta)=\frac{12}{5}$ , then the value of $\sin (\theta)=\frac{5}{12}$

In the light of the above statements, choose the correct answer form the question below:

Both Statement I and Statement II are true

Both Statement I and Statement II are false

Statement I is true but Statement II is false

Statement I is false but Statement II is true

✅ Correct Option: 3

Statement I: If $\sin(\theta) = \frac{5}{13}$

In right triangle: $\sin^2(\theta) + \cos^2(\theta) = 1$

$(\frac{5}{13})^2 + \cos^2(\theta) = 1$

$\cos^2(\theta) = 1 - \frac{25}{169} = \frac{144}{169}$

$\cos(\theta) = \frac{12}{13}$

Therefore $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{5}{13} × \frac{13}{12} = \frac{5}{12}$ ✓

Statement II: If $\cot(\theta) = \frac{12}{5}$

$\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} = \frac{12}{5}$

Therefore $\sin(\theta) = \frac{5}{13}$ (using $\sin^2(\theta) + \cos^2(\theta) = 1$ again)

Not equal to $\frac{5}{12}$ ✗

Therefore, Statement I is true and Statement II is false.

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