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QuestionFind the six trigonometric functions for angle θ\theta in a right triangle where the opposite side is 16 and adjacent side is 12.

Studdy Solution

STEP 1

Assumptions1. The triangle is a right triangle with one of the acute angles as θ\theta. . The side opposite to θ\theta is16 units.
3. The side adjacent to θ\theta is12 units.
4. We are asked to find the six trigonometric functions of θ\theta.

STEP 2

First, we need to find the length of the hypotenuse of the right triangle. We can do this by using the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Hypotenuse=Opposite2+Adjacent2Hypotenuse = \sqrt{{Opposite}^2 + {Adjacent}^2}

STEP 3

Now, plug in the given values for the opposite side and the adjacent side to calculate the length of the hypotenuse.
Hypotenuse=162+122Hypotenuse = \sqrt{{16}^2 + {12}^2}

STEP 4

Calculate the length of the hypotenuse.
Hypotenuse=162+122=256+144=400=20Hypotenuse = \sqrt{{16}^2 + {12}^2} = \sqrt{256 +144} = \sqrt{400} =20

STEP 5

Now that we have the lengths of all three sides of the triangle, we can calculate the six trigonometric functions of θ\theta.
The sine function is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.
sin⁡θ=OppositeHypotenuse\sin \theta = \frac{Opposite}{Hypotenuse}

STEP 6

Plug in the values for the opposite side and the hypotenuse to calculate sin⁡θ\sin \theta.
sin⁡θ=1620\sin \theta = \frac{16}{20}

STEP 7

implify the fraction to find sin⁡θ\sin \theta.
sin⁡θ=1620=45\sin \theta = \frac{16}{20} = \frac{4}{5}

STEP 8

The cosine function is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
cos⁡θ=AdjacentHypotenuse\cos \theta = \frac{Adjacent}{Hypotenuse}

STEP 9

Plug in the values for the adjacent side and the hypotenuse to calculate cos⁡θ\cos \theta.
cos⁡θ=1220\cos \theta = \frac{12}{20}

STEP 10

implify the fraction to find cos⁡θ\cos \theta.
cos⁡θ=1220=35\cos \theta = \frac{12}{20} = \frac{3}{5}

STEP 11

The tangent function is defined as the ratio of the sine of the angle to the cosine of the angle.
tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}

STEP 12

Plug in the values for sin⁡θ\sin \theta and cos⁡θ\cos \theta to calculate tan⁡θ\tan \theta.
tan⁡θ=4/5/5\tan \theta = \frac{4/5}{/5}

STEP 13

implify the fraction to find tan⁡θ\tan \theta.
tan⁡θ=/53/5=3\tan \theta = \frac{/5}{3/5} = \frac{}{3}

STEP 14

The cosecant function is defined as the reciprocal of the sine function.
csc⁡θ=sin⁡θ\csc \theta = \frac{}{\sin \theta}

STEP 15

Plug in the value for sin⁡θ\sin \theta to calculate csc⁡θ\csc \theta.
csc⁡θ=4/5\csc \theta = \frac{}{4/5}

STEP 16

implify the fraction to find csc⁡θ\csc \theta.
csc⁡θ=4/5=54\csc \theta = \frac{}{4/5} = \frac{5}{4}

STEP 17

The secant function is defined as the reciprocal of the cosine function.
sec⁡θ=cos⁡θ\sec \theta = \frac{}{\cos \theta}

STEP 18

Plug in the value for cos⁡θ\cos \theta to calculate sec⁡θ\sec \theta.
sec⁡θ=3/5\sec \theta = \frac{}{3/5}

STEP 19

implify the fraction to find sec⁡θ\sec \theta.
sec⁡θ=13/5=53\sec \theta = \frac{1}{3/5} = \frac{5}{3}

STEP 20

The cotangent function is defined as the reciprocal of the tangent function.
cot⁡θ=tan⁡θ\cot \theta = \frac{}{\tan \theta}

STEP 21

Plug in the value for tan⁡θ\tan \theta to calculate cot⁡θ\cot \theta.
cot⁡θ=14/3\cot \theta = \frac{1}{4/3}

STEP 22

implify the fraction to find cot⁡θ\cot \theta.
cot⁡θ=14/=4\cot \theta = \frac{1}{4/} = \frac{}{4}The six trigonometric functions of θ\theta aresin⁡θ=45\sin \theta = \frac{4}{5}cos⁡θ=5\cos \theta = \frac{}{5}tan⁡θ=4\tan \theta = \frac{4}{}csc⁡θ=54\csc \theta = \frac{5}{4}sec⁡θ=5\sec \theta = \frac{5}{}cot⁡θ=4\cot \theta = \frac{}{4}

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