Problem 2 - Precalculus

Question

The terminal side of an angle \(\theta\) in standard position passes through values of the six trigonometric functions for angle \(\theta\) $$.(8,4)$$

Step-by-Step Solution

Answer

Sine: \(\frac{\sqrt{5}}{5}\), Cosine: \(\frac{2\sqrt{5}}{5}\), Tangent: \(\frac{1}{2}\), Cosecant: \(\sqrt{5}\), Secant: \(\frac{\sqrt{5}}{2}\), Cotangent: 2.

Step 1: Understand the Situation

The terminal side of an angle \( \theta \) in standard position passes through the point (8, 4). This means the point is on the coordinate plane, and the angle is drawn from the positive x-axis to this point.

Step 2: Calculate the Radius

To calculate the trigonometric functions, we need the radius \( r \), which is the distance from the origin (0,0) to the point (8,4). Use the distance formula: \[ r = \sqrt{x^2 + y^2} = \sqrt{8^2 + 4^2} = \sqrt{64 + 16} = \sqrt{80} = 4\sqrt{5} \]

Step 3: Calculate Sine and Cosine

Using the definitions of sine and cosine: \[ \sin \theta = \frac{y}{r} = \frac{4}{4\sqrt{5}} = \frac{1}{\sqrt{5}} = \frac{\sqrt{5}}{5} \] and \[ \cos \theta = \frac{x}{r} = \frac{8}{4\sqrt{5}} = \frac{2}{\sqrt{5}} = \frac{2\sqrt{5}}{5} \]

Step 4: Calculate Tangent

Calculate tangent using sine and cosine: \[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{1}{\sqrt{5}}}{\frac{2}{\sqrt{5}}} = \frac{1}{2} \]

Step 5: Calculate Cosecant and Secant

Cosecant is the reciprocal of sine and secant is the reciprocal of cosine: \[ \csc \theta = \frac{1}{\sin \theta} = \sqrt{5} \] and \[ \sec \theta = \frac{1}{\cos \theta} = \frac{\sqrt{5}}{2} \]

Step 6: Calculate Cotangent

Cotangent is the reciprocal of tangent: \[ \cot \theta = \frac{1}{\tan \theta} = 2 \]

Key Concepts

sine

The sine function is a crucial concept in trigonometry, representing the ratio of the opposite side to the hypotenuse in a right-angle triangle. When dealing with coordinate geometry, as in this exercise, the sine of an angle \( \theta \) becomes the ratio of the \( y \)-coordinate to the radius \( r \), which is the distance from the origin to the given point. In our example, the point is \( (8, 4) \), and the radius \( r \) is calculated to be \( 4\sqrt{5} \). Thus, sine for this setup is computed as follows:

• \( \sin \theta = \frac{y}{r} = \frac{4}{4\sqrt{5}} = \frac{1}{\sqrt{5}} \)
• Rationalizing gives \( \sin \theta = \frac{\sqrt{5}}{5} \)

The value of sine, at 0.4472 approximately, helps in identifying the position of the angle's terminal side concerning the y-axis.
Understanding sine helps predict how the angle will affect various calculations including wave patterns in physics or currents in electricity.

cosine

In trigonometry, cosine complements sine by measuring the ratio of the adjacent side to the hypotenuse in a right triangle. For angles on the coordinate plane, cosine is the ratio of the x-coordinate to the radius \( r \). For our point \( (8, 4) \) and given our predicted radius \( 4\sqrt{5} \), the cosine is computed as:

• \( \cos \theta = \frac{x}{r} = \frac{8}{4\sqrt{5}} \)
• Rationalizing the expression leads to \( \cos \theta = \frac{2\sqrt{5}}{5} \)

The cosine value, around 0.8944, allows us to determine the distance along the x-axis our angle's terminal side has covered.
Cosine is vital in fields like engineering where understanding the length of a shadow or the horizontal component of a force can be calculated.

tangent

Tangent, another primary trigonometric function, assesses an angle by relating the opposite side to the adjacent side in a triangle. On coordinate planes, though, it is more defined by the ratio of sine to cosine, or equivalently, the \( y \)-coordinate divided by the \( x \)-coordinate. With our points and calculated sine and cosine:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{1}{\sqrt{5}}}{\frac{2}{\sqrt{5}}} = \frac{1}{2} \)

The tangent, valued at 0.5, indicates how steep the angle's line is. This function is routinely used in various domains like physics to determine the angle of inclination, or in architecture for slopes and angles of ramps. Understanding tangent can lend insights into how angles influence relative directions and magnitudes.