Question

If θ is an angle in standard position and its terminal side passes through the point ( 3, 2), find the exact value of tangent, thetanθ in simplest radical form.

Answer 1

To find the exact value of tangent (θ) in simplest radical form, we can use the given point (3, 2) to form a right triangle and calculate the sine and cosine of the angle. Then, we divide the sine by the cosine to find the tangent. The exact value of tangent (θ) is 2/3.

Step-by-step explanation:

To find the exact value of tangent (θ) in simplest radical form, we need to use the given information that the terminal side of the angle passes through the point (3, 2).

To do this, we can use the properties of right triangles and trigonometric functions.

First, we need to find the length of the sides of the right triangle formed by the given point and the origin (0, 0). Using the Pythagorean theorem, we get the length of the hypotenuse (h) as:

h = sqrt(3^2 + 2^2)

= sqrt(9 + 4)

= sqrt(13)

Next, we can find the sine and cosine of the angle using the ratios of the sides:

sin(θ) = 2/sqrt(13)

cos(θ) = 3/sqrt(13)

Finally, we can find the tangent of the angle by dividing the sine by the cosine:

tan(θ) = sin(θ) / cos(θ)

= (2/sqrt(13)) / (3/sqrt(13))

= 2/3