Question

find the values of sin θ, cos θ, and tan θ for the given right triangle. give the exact values. sin θ = cos θ = tan θ = 25 θ 24 resources read it

Explanation:

Step1: Find the missing side (opposite to θ)

Using Pythagorean theorem: \(a^2 + b^2 = c^2\), where \(c = 25\), \(b = 24\), find \(a\) (opposite side to θ).
\(a^2 + 24^2 = 25^2\)
\(a^2 = 25^2 - 24^2 = 625 - 576 = 49\)
\(a = \sqrt{49} = 7\)

Step2: Calculate \(\sin\theta\)

\(\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{24}{25}\) (Wait, no: opposite to θ is 24? Wait, no: θ is at the right angle's adjacent, so opposite side to θ is 24? Wait, no, the right angle: the triangle has hypotenuse 25, one leg 24, so the other leg (opposite to θ) is 7? Wait, no, let's recheck. The angle θ is at the corner with the right angle, so the sides: hypotenuse is 25, one leg (adjacent to θ) is 7? Wait, no, I messed up. Let's label: right triangle, right angle, so the two legs are 24 (horizontal) and \(a\) (vertical), hypotenuse 25. So θ is at the top corner (with the right angle), so the adjacent side to θ is the vertical leg (length \(a\)), opposite side is the horizontal leg (24), hypotenuse 25. Wait, no, let's draw: the triangle has hypotenuse 25, one leg (horizontal) 24, so the other leg (vertical) is 7. So angle θ is at the top (where the vertical leg and hypotenuse meet). So for angle θ:

• Opposite side: 24 (horizontal leg)
• Adjacent side: 7 (vertical leg)
• Hypotenuse: 25

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

Wait, I corrected: earlier mistake, the vertical leg is 7 (adjacent to θ), horizontal leg 24 (opposite to θ). So:

Step2 (corrected): Calculate \(\sin\theta\)

\(\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{24}{25}\)

Step3: Calculate \(\cos\theta\)

\(\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{7}{25}\)

Step4: Calculate \(\tan\theta\)

\(\tan\theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{24}{7}\)

Wait, no, wait: let's re-express the triangle. The right angle is between the vertical leg (length 7) and horizontal leg (length 24), hypotenuse 25. Angle θ is at the top (where vertical leg and hypotenuse meet). So:

• Opposite to θ: horizontal leg (24)
• Adjacent to θ: vertical leg (7)
• Hypotenuse: 25

Yes, so:
\(\sin\theta = \frac{24}{25}\) (opposite/hypotenuse)
\(\cos\theta = \frac{7}{25}\) (adjacent/hypotenuse)
\(\tan\theta = \frac{24}{7}\) (opposite/adjacent)

Answer:

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