Problem 6 - Basic Technical Mathematics with Calculus

Question

Find values of the trigonometric functions of the angle (in standard position) whose terminal side passes through the given points. For Exercises \(3-14,\) give answers in exact form. For Exercises 15 and \(16,\) the coordinates are approximate. $$(240,70)$$

Step-by-Step Solution

Answer

sin θ = 7/25, cos θ = 24/25, tan θ = 7/24.

Step 1: Understand the Given Point

We have a point (240, 70) which lies on the terminal side of an angle in standard position. This means the angle's initial side is along the positive x-axis and the terminal side passes through this point.

Step 2: Calculate the Radius (r)

The radius (r) is the distance from the origin (0,0) to the point (240, 70). Calculate it using the distance formula: \[ r = \sqrt{x^2 + y^2} = \sqrt{240^2 + 70^2} \] This simplifies to: \[ r = \sqrt{240^2 + 70^2} = \sqrt{57600 + 4900} = \sqrt{62500} = 250 \]

Step 3: Find the Sine (sin θ)

The sine of the angle is given by: \[ \sin(\theta) = \frac{y}{r} = \frac{70}{250} = \frac{7}{25} \]

Step 4: Find the Cosine (cos θ)

The cosine of the angle is given by: \[ \cos(\theta) = \frac{x}{r} = \frac{240}{250} = \frac{24}{25} \]

Step 5: Find the Tangent (tan θ)

The tangent of the angle is given by: \[ \tan(\theta) = \frac{y}{x} = \frac{70}{240} = \frac{7}{24} \]

Step 6: Find the Cosecant (csc θ)

The cosecant is the reciprocal of sine:\[ \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{25}{7} \]

Step 7: Find the Secant (sec θ)

The secant is the reciprocal of cosine:\[ \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{25}{24} \]

Step 8: Find the Cotangent (cot θ)

The cotangent is the reciprocal of tangent:\[ \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{24}{7} \]

Key Concepts

Standard Position

In trigonometry, an angle in "standard position" is oriented on a coordinate plane with its vertex at the origin (0,0). The initial side of the angle is fixed along the positive x-axis. As the angle measures counterclockwise, the terminal side sweeps through the coordinate plane. Given any point on this terminal side, such as (240, 70), we can ascertain more information about the angle by examining this location.

The significance of standard position is that it provides a consistent way to describe angles in relation to the Cartesian plane. This setup allows us to use the x and y coordinates of points to determine trigonometric values, like sine and cosine, associated with that angle.

In our example, the angle's terminal side passes through the point (240, 70). We can assume that the angle is measuring counterclockwise from the positive x-axis. By using these coordinates, together with the distance formula, we can calculate key trigonometric functions.

Sine and Cosine

Sine and cosine are fundamental trigonometric functions used to make sense of the relationship between an angle and the coordinates of a point on its terminal side. With our point (240, 70), the formulas for sine and cosine help us find the corresponding ratios using the point's coordinates.

• The **sine** of an angle (θ) is calculated by dividing the y-coordinate by the hypotenuse or radius (r). Using our example: ** <...>