Question

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 = \\)resourcesread it

Explanation:

Step1: Find the missing side

In a right triangle, use the Pythagorean theorem \(a^2 + b^2 = c^2\), where \(c = 25\) (hypotenuse), \(b = 24\) (one leg), and \(a\) is the other leg.
\(a^2 + 24^2 = 25^2\)
\(a^2 + 576 = 625\)
\(a^2 = 625 - 576 = 49\)
\(a = 7\) (since length is positive).

Step2: Define \(\sin\theta\), \(\cos\theta\), \(\tan\theta\)

For angle \(\theta\):

• Opposite side (to \(\theta\)): \(24\)
• Adjacent side (to \(\theta\)): \(7\)
• Hypotenuse: \(25\)

• \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{24}{25}\)
• \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{7}{25}\)
• \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}=\frac{24}{7}\)

Answer:

\(\sin\theta = \frac{24}{25}\), \(\cos\theta = \frac{7}{25}\), \(\tan\theta = \frac{24}{7}\)