Problem 24 - Algebra and Trigonometry Real Mathematics, Real People

Question

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. $$(-3,-\sqrt{7})$$

Step-by-Step Solution

Answer

The trigonometric values are \(sin = -\sqrt{7}/4\), \(cos = -3/4\), \(tan = \sqrt{7}/3\), \(csc = -4/\sqrt{7}\), \(sec = -4/3\), \(cot = 3/\sqrt{7}\)

Step 1: Find the Hypotenuse

Determine the radius (hypotenuse) of the triangle formed by the point and the origin. This can be found by applying the Pythagorean theorem, where the hypotenuse \(r\) is given by the equation: \(r = \sqrt{x^2 + y^2}\). Substituting the given coordinates \((-3, -\sqrt{7})\) leads to: \(r = \sqrt{(-3)^2 + (-\sqrt{7})^2} = \sqrt{9 + 7} = \sqrt{16} = 4\)

Step 2: Find Sine, Cosine and Tangent

The sine (sin) is defined as the ratio of 'opposite' (y-coordinate) to hypotenuse (r). The cosine (cos) as the ratio of 'adjacent' (x-coordinate) to hypotenuse (r). The tangent (tan) can be found as the ratio 'opposite' to 'adjacent' (y/x). From the given coordinates and the calculated hypotenuse, the following is found: \(sin = y/r = -\sqrt{7}/4\) , \(cos = x/r = -3/4\) , \(tan = y/x = \sqrt{7}/3\)

Step 3: Find Cosecant, Secant and Cotangent

The cosecant (csc), secant (sec), and cotangent (cot) are the respective reciprocals of sine, cosine, and tangent. Solving for these functions generates the following results: \(csc = r/y = -4/\sqrt{7}\) , \(sec = r/x = -4/3\) , \(cot = x/y = 3/\sqrt{7}\)