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Chapter 4: Problem 16

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. $$(-4,10)$$

Short Answer

Expert verified
The exact values of the six trigonometric functions are: sin=\(10/\sqrt{116}\), cos=\(-4/\sqrt{116}\), tan=\(-5/2\), csc=\(\sqrt{116}/10\), sec=\(-\sqrt{116}/4\), cot=\(-2/5\).

Step by step solution

01

Identify and Assign Coordinates

From the Cartesian coordinate (-4,10), the x-coordinate can be established as -4 and the y-coordinate as 10. By convention in a right triangle, the x-coordinate is 'Adjacent' and the y-coordinate is 'Opposite'.
02

Compute the Hypotenuse Using Pythagorean Theorem

The Hypotenuse (r) can be calculated using the Pythagorean theorem, \(r = \sqrt{x^{2} + y^{2}}\), where \(x\) is the adjacent side and \(y\) is the opposite side. Hence \(r = \sqrt{(-4)^{2} + 10^{2}} = \sqrt{116}\).
03

Calculate Trigonometric Functions

The six trigonometric functions of the angle can be calculated as follows: \n1. Sine function (sin) = Opposite/Hypotenuse = \(y/r = 10/\sqrt{116}\)\n2. Cosine function (cos) = Adjacent/Hypotenuse = \(x/r = -4/\sqrt{116}\)\n3. Tangent function (tan) = Opposite/Adjacent = \(y/x = 10/-4 = -5/2\)\n4. Cosecant function (csc) = Hypotenuse/Opposite = \(r/y = \sqrt{116}/10\)\n5. Secant function (sec) = Hypotenuse/Adjacent = \(r/x = -\sqrt{116}/4\)\n6. Cotangent function (cot) = Adjacent/Opposite = \(x/y = -4/10 = -2/5\)

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