Question

1. - / 1.5 points

find the values of \\( \sin \theta \\), \\( \cos \theta \\), and \\( \tan \theta \\) for the given right triangle. give the exact values.
\\( \sin \theta = \\)
\\( \cos \theta = \\)
\\( \tan \theta = \\)
right triangle with legs 12 (horizontal) and 5 (vertical), angle \\( \theta \\) at the left acute angle
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Explanation:

Step1: Find the hypotenuse

Using the Pythagorean theorem \(a^2 + b^2 = c^2\), where \(a = 12\), \(b = 5\). So \(c=\sqrt{12^2 + 5^2}=\sqrt{144 + 25}=\sqrt{169}=13\).

Step2: Calculate \(\sin\theta\)

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

Step3: Calculate \(\cos\theta\)

\(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{12}{13}\)

Step4: Calculate \(\tan\theta\)

\(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}=\frac{5}{12}\)

Answer:

\(\sin\theta=\frac{5}{13}\), \(\cos\theta=\frac{12}{13}\), \(\tan\theta=\frac{5}{12}\)