Question

problem 4. consider a right triangle with side lengths 2, 4 and an unknown hypotenuse length. the angle α is the angle opposite the leg of length 4. (a) sketch the right triangle with all the sides and α labeled. (b) find the hypotenuse length and then use the triangle to find all six trig functions of α.

Explanation:

Step1: Find the hypotenuse

Use the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\), where \(a = 2\), \(b=4\).
\(c=\sqrt{2^{2}+4^{2}}=\sqrt{4 + 16}=\sqrt{20}=2\sqrt{5}\)

Step2: Calculate \(\sin\alpha\)

\(\sin\alpha=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{4}{2\sqrt{5}}=\frac{2\sqrt{5}}{5}\)

Step3: Calculate \(\cos\alpha\)

\(\cos\alpha=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{2}{2\sqrt{5}}=\frac{\sqrt{5}}{5}\)

Step4: Calculate \(\tan\alpha\)

\(\tan\alpha=\frac{\text{opposite}}{\text{adjacent}}=\frac{4}{2} = 2\)

Step5: Calculate \(\csc\alpha\)

\(\csc\alpha=\frac{1}{\sin\alpha}=\frac{\sqrt{5}}{2}\)

Step6: Calculate \(\sec\alpha\)

\(\sec\alpha=\frac{1}{\cos\alpha}=\sqrt{5}\)

Step7: Calculate \(\cot\alpha\)

\(\cot\alpha=\frac{1}{\tan\alpha}=\frac{1}{2}\)

Answer:

Hypotenuse length \(c = 2\sqrt{5}\); \(\sin\alpha=\frac{2\sqrt{5}}{5}\), \(\cos\alpha=\frac{\sqrt{5}}{5}\), \(\tan\alpha = 2\), \(\csc\alpha=\frac{\sqrt{5}}{2}\), \(\sec\alpha=\sqrt{5}\), \(\cot\alpha=\frac{1}{2}\)