91Ó°ÊÓ

The coordinates of the point are given as \((3 \frac{1}{2},-7 \frac{3}{4})\), that means x-coordinate is \(3 \frac{1}{2}\) and y-coordinate is \(-7 \frac{3}{4}\). We now rewrite these fractional numbers into decimal form for simplicity, hence x equals 3.5 and y equals -7.75.

The hypotenuse \(r\) can be found by applying Pythagoras' theorem, which is \(r = \sqrt{x^2 + y^2} = \sqrt{(3.5)^2 + (-7.75)^2} = \sqrt{12.25 + 60.0625} = \sqrt{72.3125}\). Simplifying the square root, we get \(r \approx 8.5\).

Now, with the x, y and r values, we can compute the six trigonometric functions as follows:Sine (\(sin\)): It is defined as the ratio of y to r. Hence, \(sin = \frac{y}{r} = \frac{-7.75}{8.5} \approx -0.9118\).Cosine (\(cos\)): It is defined as the ratio of x to r. Hence, \(cos = \frac{x}{r} = \frac{3.5}{8.5} \approx 0.4118\).Tangent (\(tan\)): It is defined as the ratio of sine to cosine. Hence, \(tan = \frac{sin}{cos} = \frac{-0.9118}{0.4118} \approx -2.2141\).For the reciprocals of the above, we have:Cosecant (\(csc\)): It is the reciprocal of sine. Hence, \(csc = \frac{1}{sin} \approx -1.0967\).Secant (\(sec\)): It is the reciprocal of cosine. Hence, \(sec = \frac{1}{cos} \approx 2.4286\).Cotangent (\(cot\)): It is the reciprocal of tangent. Hence, \(cot = \frac{1}{tan} \approx -0.4514\).