12. Trigonometric Functions

Trigonometric Functions on Right Triangles

12. Trigonometric Functions

Trigonometric Functions on Right Triangles

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Multiple Choice

If $\tan\theta=\frac{12}{5}$ , find the values of the five other trigonometric functions. Rationalize the denominators if necessary.

$\sin\theta=\frac{12}{13},\cos\theta=\frac{5}{13},\cot\theta=\frac{5}{12},\sec\theta=\frac{13}{5},\csc\theta=\frac{13}{12}$

$\sin\theta=\frac{5}{13},\cos\theta=\frac{12}{13},\cot\theta=\frac{5}{12},\sec\theta=\frac{13}{12},\csc\theta=\frac{13}{5}$

$\sin\theta=\frac{12}{13},\cos\theta=\frac{5}{13},\cot\theta=-\frac{5}{12},\sec\theta=-\frac{13}{5},\csc\theta=-\frac{13}{12}$

$\sin\theta=\frac{5}{13},\cos\theta=\frac{12}{13},\cot\theta=-\frac{5}{12},\sec\theta=-\frac{13}{12},\csc\theta=-\frac{13}{5}$

Verified step by step guidance

Step 1: Recall the definition of tangent in terms of a right triangle. Tangent is defined as the ratio of the opposite side to the adjacent side: tan(θ) = opposite/adjacent. From the problem, tan(θ) = 12/5, so the opposite side is 12 and the adjacent side is 5.

Step 2: Use the Pythagorean theorem to find the hypotenuse of the triangle. The Pythagorean theorem states that hypotenuse² = opposite² + adjacent². Substituting the values, hypotenuse² = 12² + 5². Solve for the hypotenuse.

Step 3: Once the hypotenuse is found, calculate sin(θ) and cos(θ). Sine is defined as sin(θ) = opposite/hypotenuse, and cosine is defined as cos(θ) = adjacent/hypotenuse. Substitute the values of the opposite, adjacent, and hypotenuse to find sin(θ) and cos(θ).

Step 4: Use the reciprocal identities to find the remaining trigonometric functions. The reciprocal of sine is cosecant (csc(θ) = 1/sin(θ)), the reciprocal of cosine is secant (sec(θ) = 1/cos(θ)), and the reciprocal of tangent is cotangent (cot(θ) = 1/tan(θ)). Substitute the values of sin(θ), cos(θ), and tan(θ) to find csc(θ), sec(θ), and cot(θ).

Step 5: Simplify all fractions and rationalize the denominators if necessary. Ensure that all trigonometric function values are expressed in their simplest form, and confirm that the signs of the functions are consistent with the quadrant in which θ lies.