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Math  /  Trigonometry

Question1he point A(−40,−9)A(-40,-9) lies on the terminal arm of an angle in standard position. Determine the exact expression for he six trigonometric ratios of the angle.

Studdy Solution

STEP 1

1. The point A(−40,−9) A(-40, -9) is on the terminal arm of an angle in standard position.
2. The angle is measured from the positive x-axis.
3. The six trigonometric ratios are sine, cosine, tangent, cosecant, secant, and cotangent.

STEP 2

1. Calculate the radius (distance from the origin to the point).
2. Determine the sine, cosine, and tangent ratios.
3. Determine the cosecant, secant, and cotangent ratios.

STEP 3

Calculate the radius using the distance formula:
r=x2+y2 r = \sqrt{x^2 + y^2} =(−40)2+(−9)2 = \sqrt{(-40)^2 + (-9)^2} =1600+81 = \sqrt{1600 + 81} =1681 = \sqrt{1681} =41 = 41

STEP 4

Determine the sine, cosine, and tangent ratios:
Sine: sin⁡θ=yr=−941 \sin \theta = \frac{y}{r} = \frac{-9}{41}
Cosine: cos⁡θ=xr=−4041 \cos \theta = \frac{x}{r} = \frac{-40}{41}
Tangent: tan⁡θ=yx=−9−40=940 \tan \theta = \frac{y}{x} = \frac{-9}{-40} = \frac{9}{40}

STEP 5

Determine the cosecant, secant, and cotangent ratios:
Cosecant: csc⁡θ=1sin⁡θ=41−9=−419 \csc \theta = \frac{1}{\sin \theta} = \frac{41}{-9} = -\frac{41}{9}
Secant: sec⁡θ=1cos⁡θ=41−40=−4140 \sec \theta = \frac{1}{\cos \theta} = \frac{41}{-40} = -\frac{41}{40}
Cotangent: cot⁡θ=1tan⁡θ=409 \cot \theta = \frac{1}{\tan \theta} = \frac{40}{9}
The exact expressions for the six trigonometric ratios are:
sin⁡θ=−941,cos⁡θ=−4041,tan⁡θ=940 \sin \theta = -\frac{9}{41}, \quad \cos \theta = -\frac{40}{41}, \quad \tan \theta = \frac{9}{40}
csc⁡θ=−419,sec⁡θ=−4140,cot⁡θ=409 \csc \theta = -\frac{41}{9}, \quad \sec \theta = -\frac{41}{40}, \quad \cot \theta = \frac{40}{9}

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