Answer
$\sin {\theta} = \frac{\sqrt {2}}{2}$
$\cos {\theta} = \frac{\sqrt {2}}{2}$
$\tan {\theta} = 1$
$\cot {\theta} = 1$
$\sec {\theta} = \sqrt {2}$
$\sec {\theta} = \sqrt {2}$
Work Step by Step
First, we need to find the value of the hypotenuse using the Pythagorean theorem, which states that $a^2 + b^2 = c^2$, where $a$ is the length of the side adjacent to the given angle, $b$ is the length of the side opposite the angle, and $c$ is the length of the hypotenuse.
In this triangle, $a$ is $3$ and $b$ is $3$, so let's plug these values into the formula to find the length of the hypotenuse:
$3^2 + 3^2 = c^2$
Evaluate the exponents:
$9 + 9= c^2$
Add on the left side of the equation:
$c^2 = 18$
Take the square root of both sides of the equation:
$c = 3\sqrt {2}$
Now that we have the lengths of all sides of the right triangle, let's plug in the values into the formulas for the trigonometric functions:
$\sin {\theta} = \frac{opposite}{hypotenuse}$ = $\frac{3}{3\sqrt {2}}$
Simplify the fraction:
$\sin {\theta} = \frac{opposite}{hypotenuse}$ = $\frac{1}{\sqrt {2}}$
We can't leave a radical in the denominator, so we multiply both the denominator and numerator by $\sqrt {2}$:
$\sin {\theta} = \frac{opposite}{hypotenuse}$ = $\frac{\sqrt {2}}{2}$
$\cos {\theta} = \frac{opposite}{hypotenuse}$ = $\frac{3}{3\sqrt {2}}$
Simplify the fraction:
$\cos {\theta} = \frac{opposite}{hypotenuse}$ = $\frac{1}{\sqrt {2}}$
We can't leave a radical in the denominator, so we multiply both the denominator and numerator by $\sqrt {2}$:
$\cos {\theta} = \frac{opposite}{hypotenuse}$ = $\frac{\sqrt {2}}{2}$
$\tan {\theta} = \frac{opposite}{adjacent}$ = $\frac{3}{3}$ = $1$
$\cot {\theta} = \frac{adjacent}{opposite}$ = $\frac{3}{3}$ = $1$
$\sec {\theta} = \frac{hypotenuse}{adjacent}$ = $\frac{3\sqrt {2}}{3}$
Simplify the fraction:
$\sec {\theta} = \frac{hypotenuse}{adjacent}$ = $\sqrt {2}$
$\csc {\theta} = \frac{hypotenuse}{opposite}$ = $\frac{3\sqrt {2}}{3}$
Simplify the fraction:
$\sec {\theta} = \frac{hypotenuse}{adjacent}$ = $\sqrt {2}$