Question

1. for the following triangle, state the 3 primary trig ratios in relation to angle a.4

Explanation:

Step1: Recall trig - ratio definitions

The three primary trigonometric ratios are sine ($\sin$), cosine ($\cos$), and tangent ($\tan$). For a right - triangle with respect to an angle $A$, $\sin A=\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos A = \frac{\text{adjacent}}{\text{hypotenuse}}$, and $\tan A=\frac{\text{opposite}}{\text{adjacent}}$.

Step2: Identify sides

In the given right - triangle, the side opposite to angle $A$ has length $5$ (using the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where $c = 13$ and $b = 12$, so $a=\sqrt{13^{2}-12^{2}}=\sqrt{169 - 144}=\sqrt{25}=5$), the side adjacent to angle $A$ has length $12$, and the hypotenuse has length $13$.

Step3: Calculate $\sin A$

$\sin A=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{5}{13}$

Step4: Calculate $\cos A$

$\cos A=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{12}{13}$

Step5: Calculate $\tan A$

$\tan A=\frac{\text{opposite}}{\text{adjacent}}=\frac{5}{12}$

Answer:

$\sin A=\frac{5}{13}$, $\cos A=\frac{12}{13}$, $\tan A=\frac{5}{12}$