Problem 4 - Precalculus Essentials

Question

In Exercises \(1-8,\) a point on the terminal side of angle \(\theta\) is given. Find the exact value of each of the six trigonometric functions of \(\theta\). $$(3,7)$$

Step-by-Step Solution

Answer

The six trigonometric values for the point (3,7) can be expressed as follows: sin(θ)=7/r, cos(θ)=3/r, tan(θ)=7/3, csc(θ)=r/7, sec(θ)=r/3, and cot(θ)=3/7 where r is the angle's distance found to be \(\sqrt{3^2 + 7^2}\)

Step 1: Determine r(Angle's Distance)

First, determine the value for r, the distance from the origin to the point (3,7) on the angle's terminal side. This is done using the Pythagorean theorem: \(r = \sqrt{x^2 + y^2}\) where x and y are the coordinates of the point (3,7). Here, \(r = \sqrt{3^2 + 7^2}\)

Step 2: Calculate the Sine Value

Sine (sin) of θ can be calculated as the y coordinate divided by r. Here, \(sin(θ) = \frac{y}{r} = \frac{7}{r}\)

Step 3: Calculate the Cosine Value

Cosine (cos) of θ can be calculated as the x coordinate divided by r. Here, \(cos(θ) = \frac{x}{r} = \frac{3}{r}\)

Step 4: Calculate the Tangent Value

Tangent (tan) of θ is the ratio of sin(θ) to cos(θ). Hence, \(tan(θ) = \frac{sin(θ)}{cos(θ)} = \frac{7/3}\)

Step 5: Calculate Cosecant, Secant and Cotangent Values

Cosecant (csc), secant (sec) and cotangent (cot) values are the reciprocals of sin, cos and tan respectively. Hence, \(csc(θ) = \frac{1}{sin(θ)} = \frac{r}{7}\), \(sec(θ) = \frac{1}{cos(θ)} = \frac{r}{3}\) and \(cot(θ) = \frac{1}{tan(θ)} = \frac{3}{7}\)