Give that tan o =3/4 Fine a sim b cos c7tan + 5 cos

Question

Give that tan o =3/4 Fine a sim b cos c7tan + 5 cos

Asked on March 29, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-03-29T16:07:10.305Z

Given $\tan \theta = \frac{3}{4}$. Step 1: Find $\sin \theta$ and $\cos \theta$. We can use a right-angled triangle where $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$. Let the opposite side be $3k$ and the adjacent side be $4k$ for some constant $k$. Using the Pythagorean theorem, the hypotenuse $h$ is: $$h = \sqrt{(3k)^2 + (4k)^2} = \sqrt{9k^2 + 16k^2} = \sqrt{25k^2} = 5k$$ Now we can find $\sin \theta$ and $\cos \theta$: $$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3k}{5k} = \frac{3}{5}$$ $$\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{4k}{5k} = \frac{4}{5}$$ Step 2: Evaluate the expression $7 \tan \theta + 5 \cos \theta$. Substitute the given value of $\tan \theta$ and the calculated value of $\cos \theta$: $$7 \tan \theta + 5 \cos \theta = 7 \left(\frac{3}{4}\right) + 5 \left(\frac{4}{5}\right)$$ $$= \frac{21}{4} + \frac{20}{5}$$ $$= \frac{21}{4} + 4$$ To add these fractions, find a common denominator: $$= \frac{21}{4} + \frac{4 \times 4}{4}$$ $$= \frac{21}{4} + \frac{16}{4}$$ $$= \frac{21 + 16}{4}$$ $$= \frac{37}{4}$$ The value of the expression is $\boxed{\frac{37}{4}}$.

Accepted Answer · 2026-03-29T16:07:10.305Z

Given $\tan \theta = \frac{3}{4}$. Step 1: Find $\sin \theta$ and $\cos \theta$. We can use a right-angled triangle where $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$. Let the opposite side be $3k$ and the adjacent side be $4k$ for some constant $k$. Using the Pythagorean theorem, the hypotenuse $h$ is: $$h = \sqrt{(3k)^2 + (4k)^2} = \sqrt{9k^2 + 16k^2} = \sqrt{25k^2} = 5k$$ Now we can find $\sin \theta$ and $\cos \theta$: $$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3k}{5k} = \frac{3}{5}$$ $$\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{4k}{5k} = \frac{4}{5}$$ Step 2: Evaluate the expression $7 \tan \theta + 5 \cos \theta$. Substitute the given value of $\tan \theta$ and the calculated value of $\cos \theta$: $$7 \tan \theta + 5 \cos \theta = 7 \left(\frac{3}{4}\right) + 5 \left(\frac{4}{5}\right)$$ $$= \frac{21}{4} + \frac{20}{5}$$ $$= \frac{21}{4} + 4$$ To add these fractions, find a common denominator: $$= \frac{21}{4} + \frac{4 \times 4}{4}$$ $$= \frac{21}{4} + \frac{16}{4}$$ $$= \frac{21 + 16}{4}$$ $$= \frac{37}{4}$$ The value of the expression is $\boxed{\frac{37}{4}}$.