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Chapter 4: Problem 17

The point is on the terminal side of an angle in standard position. Determine the exact values of the six trigonometric functions of the angle. $$(-5.4,7.2)$$

Short Answer

Expert verified
The six trigonometric functions of the angle are: \(cos = -0.6\), \(sin = 0.8\), \(tan = -4/3\), \(sec = -5/3\), \(csc = 5/4\), \(cot = -3/4\)

Step by step solution

01

Determine the values of cosine and sine.

The cosine (abbreviated as cos) of an angle in standard position is defined as the x-coordinate of a point on the terminal side of the angle, divided by the distance from the origin to the point. Meanwhile, the sine (abbreviated as sin) is defined as the y-coordinate of the point divided by the same distance. Given the point is (-5.4,7.2), you can immediately determine the coordinates. Note that in the second quadrant, cosine is negative and sine is positive. The distance from the origin to the point can be calculated using the Pythagorean Theorem, which gives \(\sqrt{(-5.4)^2+(7.2)^2}\). So, \(cos = -5.4/\sqrt{(-5.4)^2+(7.2)^2}\) and \(sin = 7.2/\sqrt{(-5.4)^2+(7.2)^2}\)
02

Calculate the values of cosine and sine.

Compute the values. The results are: \(cos = -5.4/\sqrt{(-5.4)^2+(7.2)^2} = -0.6\) and \(sin = 7.2/\sqrt{(-5.4)^2+(7.2)^2} = 0.8\)
03

Determine the values of the rest trigonometric functions.

The remaining trigonometric functions can be expressed in terms of sine and cosine. The tangent (abbreviated as tan) is sin divided by cos. The secant (abbreviated as sec) is the reciprocal of cosine. The cosecant (abbreviated as csc) is the reciprocal of sine. The cotangent (abbreviated as cot) is the reciprocal of tangent. So, we have: \(tan = sin/cos = 0.8/(-0.6)\), \(sec = 1/cos = 1/(-0.6)\), \(csc = 1/sin = 1/0.8\), \(cot = 1/tan = -0.6/0.8\)
04

Calculate the values of the rest trigonometric functions.

Compute the values for tangent, secant, cosecant, and cotangent. The results are: \(tan = sin/cos = 0.8/(-0.6) = -4/3\), \(sec = 1/cos = 1/(-0.6) = -5/3\), \(csc = 1/sin = 1/0.8 = 5/4\), \(cot = 1/tan = -0.6/0.8 = -3/4\)

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