Problem 11 - Precalculus with Limits

Question

In Exercises 11 - 24, use the given values to evaluate (if possible)all six trigonometric functions. \( \sin x = \dfrac{1}{2} \), \( \cos x = \dfrac{\sqrt{3}}{2} \)

Step-by-Step Solution

Answer

The six evaluated trigonometric functions are: \( \sin x = \dfrac{1}{2} \), \( \cos x = \dfrac{\sqrt{3}}{2} \), \( \tan x = \dfrac{1}{\sqrt{3}} \), \( \csc x = 2 \), \( \sec x = \dfrac{2}{\sqrt{3}} \), and \( \cot x = \sqrt{3} \).

Step 1: Evaluate the Given Functions

The values for \( \sin x \) and \( \cos x \) are already provided: \( \sin x = \dfrac{1}{2} \) and \( \cos x = \dfrac{\sqrt{3}}{2} \).

Step 2: Evaluate the Remaining Trigonometric Functions

The remaining trigonometric functions can be evaluated based on the given values. Here are the expressions for them: \n\n\t- Tangent function: \( \tan x = \dfrac{\sin x}{\cos x} = \dfrac{\dfrac{1}{2}}{\dfrac{\sqrt{3}}{2}} = \dfrac{1}{\sqrt{3}} \)\n\n\t- Cosecant function: \( \csc x = \dfrac{1}{\sin x} = \dfrac{1}{\dfrac{1}{2}} = 2 \)\n\n\t- Secant function: \( \sec x = \dfrac{1}{\cos x} = \dfrac{1}{\dfrac{\sqrt{3}}{2}} = \dfrac{2}{\sqrt{3}} \)\n\n\t- Cotangent function: \( \cot x = \dfrac{1}{\tan x} = \dfrac{1}{\dfrac{1}{\sqrt{3}}} = \sqrt{3} \)

Key Concepts

sine function

The sine function is one of the primary trigonometric functions. It is denoted as \( \sin \theta \), where \( \theta \) is an angle in a right triangle.
This function helps in determining the ratio of the length of the side of the triangle opposite to the angle to the length of the hypotenuse.
For example, if given \( \sin x = \frac{1}{2} \), it means the opposite side is half the length of the hypotenuse.

• The sine function is periodic, with a period of \( 2\pi \). This periodic nature allows it to repeat its values in cycles.
• Its range is between -1 and 1, inclusive, as it measures ratio based on triangle sides.

Sine functions are often used to model wave-like phenomena, connecting trigonometry to real-world physics.

cosine function

The cosine function, noted as \( \cos \theta \), complements the sine function. In a right triangle, it represents the ratio of the length of the adjacent side to the hypotenuse.
For instance, \( \cos x = \frac{\sqrt{3}}{2} \), suggests that the adjacent side is \( \frac{\sqrt{3}} \) times as large as half the hypotenuse.

• The cosine function is also periodic with a period of \( 2\pi \).
• Its range is identical to that of the sine function, going from -1 to 1.

This function can be visualized as the horizontal distance from a point on a unit circle to the vertical axis, aiding in understanding angles and distances in trigonometry.

tangent function

Tangent, denoted as \( \tan \theta \), is another essential trigonometric function. It expresses the ratio of sine to cosine, or more simply, the opposite to adjacent side in a right triangle.
In our exercise, \( \tan x = \frac{1}{\sqrt{3}} \), which can be thought of as the slope of the line connecting the point to the origin in a triangle.

• Tangent is periodic with a period of \( \pi \), making it different from the sine and cosine functions in terms of repetition.
• Unlike sine and cosine, the range of tangent is all real numbers from negative to positive infinity.

It helps in calculating angles and is particularly useful in understanding phenomena that involve directional changes, like slopes and inclines.

cosecant function

The cosecant function, represented as \( \csc \theta \), is the reciprocal of the sine function.
Its formula is easy: \( \csc x = \frac{1}{\sin x} \). In the example, it equals 2, indicating that the hypotenuse is twice as long as the opposite side.

• Cosecant does not include zero in its range since that would result in division by zero.
• Its range excludes values between -1 and 1 but spans from negative to positive infinity outside that interval.

Though often less emphasized, the cosecant function is crucial in contexts where ratios are used inversely, such as certain optical phenomena or waves.

secant function

The secant function is the reciprocal of the cosine function. It is denoted as \( \sec \theta \), calculated through \( \frac{1}{\cos \theta} \).
For \( \cos x = \frac{\sqrt{3}}{2} \), the secant function results in \( \frac{2}{\sqrt{3}} \).

• Similar to cosecant, the secant excludes zero in its range, diverging at the points where cosine equals zero.
• Its range also excludes values from -1 to 1, extending into infinities in both directions.

The secant's role is indispensable in engineering and physics, especially when dealing with wave properties or rotational angles.

cotangent function

Cotangent is the inverse of the tangent function, denoted \( \cot \theta \). It is calculated as \( \frac{1}{\tan \theta} \), or the adjacent over the opposite side in a triangle.
In the provided example, \( \cot x = \sqrt{3} \), representing a particular angle's inclination.

• Like tangent, cotangent is periodic with a period of \( \pi \).
• Its range covers the entire real number line, indicating how it can represent various slopes and angles.

Cotangent finds its use in mechanical contexts, signal processing, and computations requiring fine angle adjustments.