Answer
sin θ = $\frac{opposite}{hypotenuse}$ = $\frac{3}{5}$
cos θ = $\frac{adjacent}{hypotenuse}$ = $\frac{4}{5}$
tan θ = $\frac{opposite}{adjacent}$ = $\frac{3}{4}$
cot θ = $\frac{adjacent}{opposite}$ = $\frac{4}{3}$
sec θ = $\frac{hypotenuse}{adjacent}$ = $\frac{5}{4}$
csc θ = $\frac{hypotenuse}{opposite}$ = $\frac{5}{3}$
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 $4$ and $b$ is $3$, so let's plug these values into the formula to find the length of the hypotenuse:
$4^2 + 3^2 = c^2$
Evaluate the exponents:
$16 + 9 = c^2$
Add on the left side of the equation:
$c^2 = 25$
Take the square root of both sides of the equation:
$c = 5$
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 θ = $\frac{opposite}{hypotenuse}$ = $\frac{3}{5}$
cos θ = $\frac{adjacent}{hypotenuse}$ = $\frac{4}{5}$
tan θ = $\frac{opposite}{adjacent}$ = $\frac{3}{4}$
cot θ = $\frac{adjacent}{opposite}$ = $\frac{4}{3}$
sec θ = $\frac{hypotenuse}{adjacent}$ = $\frac{5}{4}$
csc θ = $\frac{hypotenuse}{opposite}$ = $\frac{5}{3}$
