Problem 74 - A Graphical Approach to Precalculus with Limits

Question

Verify that equation is an identity. \(\frac{\tan \theta}{\sec \theta}=\sin \theta\)

Step-by-Step Solution

Answer

The equation \(\frac{\tan \theta}{\sec \theta} = \sin \theta\) is an identity.

Step 1: Express All Trigonometric Functions in Terms of Sine and Cosine

The identity we need to verify is \(\frac{\tan \theta}{\sec \theta} = \sin \theta\). Start by expressing \(\tan \theta\) and \(\sec \theta\) in terms of \(\sin \theta\) and \(\cos \theta\). The tangent of \(\theta\) is \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) and the secant of \(\theta\) is \(\sec \theta = \frac{1}{\cos \theta}\). Substituting these, the left side becomes: \(\frac{\frac{\sin \theta}{\cos \theta}}{\frac{1}{\cos \theta}}\).

Step 2: Simplify the Expression

Simplify the expression from Step 1. Since division by a fraction is the same as multiplying by its reciprocal, we get \(\frac{\sin \theta}{\cos \theta} \times \cos \theta\). This simplifies to just \(\sin \theta\), because the \(\cos \theta\) in the numerator and denominator cancel out.

Step 3: Confirm Left Side Equals Right Side

Now compare the simplified version of the left side (\(\sin \theta\)) to the right side of the original identity, which is also \(\sin \theta\). Both sides are equal, confirming that the original equation \(\frac{\tan \theta}{\sec \theta} = \sin \theta\) is indeed an identity.

Key Concepts

Sine Function

The sine function is one of the primary trigonometric functions. It is represented by the symbol \(\sin \theta\) and is defined as the ratio between the length of the opposite side and the hypotenuse of a right-angled triangle. This means that for a given angle \(\theta\):

• \(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}\)

Using the unit circle, the sine of an angle provides the y-coordinate of the corresponding point on the circle. This makes it a fundamental aspect of understanding wave patterns, especially in physics and engineering.
The sine function is periodic with a period of \(2\pi\), meaning it repeats its values every \(2\pi\) radians. Understanding the sine function is crucial as it helps simplify complex trigonometric identities by relating other functions like tangent and secant back to sine and cosine.
When verifying trigonometric identities, expressing everything in terms of sine and cosine is a helpful strategy. This makes equations simpler and often more intuitive.

Cosine Function

The cosine function, represented by \(cos \theta\), is another essential trigonometric function. It measures the ratio of the adjacent side to the hypotenuse in a right-angled triangle:

• \(cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}\)

On the unit circle, the cosine of an angle is the x-coordinate of the corresponding point. Just like sine, cosine is periodic with a period of \(2\pi\).The cosine function is not only important in forming basic trigonometric identities but is also used in many areas such as signal processing and oscillatory systems. In problems like the given identity \(\frac{\tan \theta}{\sec \theta} = \sin \theta\), the cosine is important because both the tangent and secant functions are directly derivable from it, highlighting its substantial role in simplifying and solving these identities.
By expressing tangent and secant in terms of sine and cosine, we can unravel complex trigonometric equations more efficiently.

Tangent Function

The tangent function \(\tan \theta\) is a prominent trigonometric function, defined as the ratio of the sine to the cosine of an angle:

• \(tan \theta = \frac{\sin \theta}{\cos \theta}\)

In the context of right-angled triangles, tangent represents the ratio of the opposite side to the adjacent side. On the unit circle, this is visualized as the slope of a line representing a given angle.
Tangent has periodic behavior with a period of \(\pi\), which means it repeats its pattern every \(\pi\) radians. Understanding how to manipulate the tangent function is key in verifying complex identities such as \(\frac{\tan \theta}{\sec \theta} = \sin \theta\). This identity can be broken down by expressing tangent and secant in terms of sine and cosine, greatly simplifying the verification process. In any trigonometric identity problem, breaking down the tangent and related functions into simpler components helps in recognizing patterns or establishing equality between different expressions.