Given that $\tan \theta = \frac{1}{3}$ and $\sin \theta > 0$ , what is $\cos \theta$ ?

Full solution

Q. Given that

\tan \theta = \frac{1}{3}

and

\sin \theta > 0

, what is

\cos \theta

Given Information: We are given that $\tan \theta = \frac{1}{3}$ and $\sin \theta > 0$ . We need to find $\cos \theta$ . The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. Since $\tan \theta = \frac{1}{3}$ , we can consider the opposite side to be $1$ and the adjacent side to be $3$ in a right triangle.
Finding Opposite and Adjacent Sides: Using the Pythagorean theorem, we can find the hypotenuse of the triangle. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. $\newline$ Let's denote the hypotenuse as $h$ . Then we have: $\newline$ $h^2 = \text{opposite}^2 + \text{adjacent}^2$ $\newline$ $h^2 = 1^2 + 3^2$ $\newline$ $h^2 = 1 + 9$ $\newline$ $h^2 = 10$ $\newline$ $h = \sqrt{10}$
Finding Hypotenuse: Now that we have the hypotenuse, we can find $\cos \theta$ . The cosine of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse. $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$ $\cos \theta = \frac{3}{\sqrt{10}}$
Finding Cosine: To rationalize the denominator, we multiply the numerator and the denominator by $\sqrt{10}$ .
$\cos \theta = \frac{3}{\sqrt{10}} \times \frac{\sqrt{10}}{\sqrt{10}}$
$\cos \theta = \frac{3\sqrt{10}}{10}$