Question

trig #1
write an equation to find the missing angle.

Explanation:

Step1: Recall trigonometric ratios

In a right - triangle, \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\)

Step2: Identify sides relative to \(\theta\)

For angle \(\theta\), the adjacent side is \(12\) (side \(AC\)), the hypotenuse is \(13\) (side \(AB\)), and the opposite side is \(BC=\sqrt{13^{2}-12^{2}} = 5\) (using the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c = 13\) and \(a = 12\), then \(b=\sqrt{c^{2}-a^{2}}\))

Step3: Check each option

• For \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{5}{13}

eq\frac{12}{13}\)

• For \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{12}{13}\)
• For \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}=\frac{5}{12}

eq\frac{12}{13}\) and \(\tan\theta
eq\frac{13}{12}\)

Answer:

\(\cos\theta=\frac{12}{13}\) (the second option)