Question

3.) directions: find the trig functions

sina=_____ sinb=_____
cosa=_____ cosb=_____
tana=_____ tanb=_____
4.) determine which trig ratio would be used to solve for the variable.
____(__)°= ____
3.) in △abc, ab = 20, bc = 21, and ca = 29, where ∠b is a right angle. what is the value of tan(a)?

Explanation:

First Problem (Right Triangle with legs 5, 12)

Step1: Calculate hypotenuse

Use Pythagorean theorem:
$$c = \sqrt{5^2 + 12^2} = \sqrt{25+144} = \sqrt{169} = 13$$

Step2: Find sinA (opp/hyp)

$\sin A = \frac{\text{opposite to } A}{\text{hypotenuse}} = \frac{12}{13}$

Step3: Find sinB (opp/hyp)

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

Step4: Find cosA (adj/hyp)

$\cos A = \frac{\text{adjacent to } A}{\text{hypotenuse}} = \frac{5}{13}$

Step5: Find cosB (adj/hyp)

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

Step6: Find tanA (opp/adj)

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

Step7: Find tanB (opp/adj)

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

Step1: Identify sides relative to 37°

$x$ is adjacent, 19 is opposite to 37°.

Step2: Select correct trig ratio

Use tangent (opp/adj):
$$\tan(37^\circ) = \frac{19}{x}$$

Step1: Identify sides for $\tan(A)$

Right angle at $B$, so:
Opposite to $A$ = $BC=21$, Adjacent to $A$ = $AB=20$

Step2: Calculate $\tan(A)$

$\tan(A) = \frac{\text{opposite}}{\text{adjacent}} = \frac{21}{20}$

Answer:

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

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Second Problem (Solve for $x$)