Questions: Each point listed is on terminal side of an angle. Show that each of the indicated functions is the same for each of the points (5,12), (10,24), (2.5,6), sin θ and tan θ Find the values of sin θ and tan θ for the point (5,12). sin θ= (Simplify your answers. Type integers or fractions.)

Solution Steps

To find the values of \(\sin \theta\) and \(\tan \theta\) for the point \((5, 12)\), we need to use the definitions of sine and tangent in terms of the coordinates of the point on the terminal side of the angle. Specifically, we use the following formulas:

• \(\sin \theta = \frac{y}{r}\)
• \(\tan \theta = \frac{y}{x}\)

where \(x\) and \(y\) are the coordinates of the point, and \(r\) is the distance from the origin to the point, calculated as \(r = \sqrt{x^2 + y^2}\).

To find \( r \), we use the formula: \[ r = \sqrt{x^2 + y^2} \] Substituting \( x = 5 \) and \( y = 12 \): \[ r = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \]

Using the definition of sine: \[ \sin \theta = \frac{y}{r} \] Substituting the values: \[ \sin \theta = \frac{12}{13} \approx 0.9231 \]

Using the definition of tangent: \[ \tan \theta = \frac{y}{x} \] Substituting the values: \[ \tan \theta = \frac{12}{5} = 2.4 \]

Final Answer