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Sketch an angle
theta in standard position such that
theta has the least possible positive measure, and the point
(-7,-24) is on the terminal side of
theta. Then find the values of the six trigonometric functions for the angle. Rationalize denominators when applicable. Do not use a calculator.

Sketch an angle $\theta$ in standard position such that $\theta$ has the least possible positive measure, and the point $(-7,-24)$ is on the terminal side of $\theta$ . Then find the values of the six trigonometric functions for the angle. Rationalize denominators when applicable. Do not use a calculator.

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Find the constant of variation

Full solution

Q. Sketch an angle

\theta

in standard position such that

\theta

has the least possible positive measure, and the point

(-7,-24)

is on the terminal side of

\theta

. Then find the values of the six trigonometric functions for the angle. Rationalize denominators when applicable. Do not use a calculator.

Determine Quadrant: First, we need to determine the quadrant in which the angle $\theta$ 's terminal side lies since the point $(-7, -24)$ has both negative $x$ and $y$ coordinates, $\theta$ is in the third quadrant.
Find Reference Angle: To find the reference angle, we use the coordinates $(-7, -24)$ . The reference angle is calculated using the tangent, which is the ratio of the y-coordinate to the x-coordinate. So, $\tan(\text{reference angle}) = \frac{24}{7}$ .
Calculate Trigonometric Functions: Now, we calculate the six trigonometric functions for $\theta$ . Since $\theta$ is in the third quadrant, sine and cosine are negative, and tangent is positive. We start by finding the hypotenuse using the Pythagorean theorem: $\text{hypotenuse} = \sqrt{(-7)^2 + (-24)^2} = \sqrt{49 + 576} = \sqrt{625} = 25$ .
Find Hypotenuse: The sine of theta is the y-coordinate divided by the hypotenuse, so $\sin(\theta) = -\frac{24}{25}$ . The cosine of theta is the x-coordinate divided by the hypotenuse, so $\cos(\theta) = -\frac{7}{25}$ . The tangent of theta is the y-coordinate divided by the x-coordinate, so $\tan(\theta) = \frac{24}{7}$ .
Reciprocal Trigonometric Functions: For the reciprocal trigonometric functions, cosecant is the reciprocal of sine, so $\csc(\theta) = -\frac{25}{24}$ . Secant is the reciprocal of cosine, so $\sec(\theta) = -\frac{25}{7}$ . Cotangent is the reciprocal of tangent, so $\cot(\theta) = \frac{7}{24}$ .

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