Question

assume that θ is an acute angle in a right - triangle satisfying the given condition. evaluate the remaining trigonometric functions. tan θ = 5/9 sin θ = □ (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Recall tangent - side relationship

In a right - triangle, $\tan\theta=\frac{opposite}{adjacent}$. Given $\tan\theta = \frac{5}{9}$, let the opposite side $a = 5$ and the adjacent side $b = 9$.

Step2: Find the hypotenuse $c$

By the Pythagorean theorem $c=\sqrt{a^{2}+b^{2}}$. So $c=\sqrt{5^{2}+9^{2}}=\sqrt{25 + 81}=\sqrt{106}$.

Step3: Calculate $\sin\theta$

$\sin\theta=\frac{opposite}{hypotenuse}=\frac{5}{\sqrt{106}}=\frac{5\sqrt{106}}{106}$.

Step4: Calculate $\cos\theta$

$\cos\theta=\frac{adjacent}{hypotenuse}=\frac{9}{\sqrt{106}}=\frac{9\sqrt{106}}{106}$.

Step5: Calculate $\csc\theta$

$\csc\theta=\frac{1}{\sin\theta}=\frac{\sqrt{106}}{5}$.

Step6: Calculate $\sec\theta$

$\sec\theta=\frac{1}{\cos\theta}=\frac{\sqrt{106}}{9}$.

Step7: Calculate $\cot\theta$

$\cot\theta=\frac{1}{\tan\theta}=\frac{9}{5}$.

Answer:

• $\sin(\theta)=\frac{5\sqrt{106}}{106}$
• $\cos(\theta)=\frac{9\sqrt{106}}{106}$
• $\tan(\theta)=\frac{5}{9}$
• $\csc(\theta)=\frac{\sqrt{106}}{5}$
• $\sec(\theta)=\frac{\sqrt{106}}{9}$
• $\cot(\theta)=\frac{9}{5}$