Problem 5 - Algebra and Trigonometry

Question

Write the trigonometric expression in terms of sine and cosine, and then simplify. $$ \sin \theta \sec \theta $$

Step-by-Step Solution

Answer

The expression simplifies to \( \tan \theta \).

Step 1: Express Secant in Terms of Cosine

The secant function, \( \sec \theta \), can be expressed in terms of cosine as \( \sec \theta = \frac{1}{\cos \theta} \). Substitute this into the given expression.

Step 2: Substitute the Expression

Substitute the expression for secant into the original expression: \[\sin \theta \sec \theta = \sin \theta \times \frac{1}{\cos \theta}\].

Step 3: Simplify the Expression

Simplify the expression by performing the multiplication: \[\sin \theta \times \frac{1}{\cos \theta} = \frac{\sin \theta}{\cos \theta}\].

Step 4: Recognize the Simplified Expression

The expression \( \frac{\sin \theta}{\cos \theta} \) is known as \( \tan \theta \), the tangent of \( \theta \). Thus, the simplified expression is \( \tan \theta \).

Key Concepts

Understanding the Sine Function

The sine function is a fundamental aspect of trigonometry, representing the relationship between the angle in a right triangle and the ratio of the opposite side to the hypotenuse. It's denoted as \( \sin \theta \). The sine function is periodic, meaning it repeats its values in regular intervals. Specifically, the sine function completes a full cycle every \( 2\pi \) radians or 360 degrees.

• The sine of 0 degrees or 0 radians is 0.
• The sine of 90 degrees or \( \frac{\pi}{2} \) radians is 1.
• It smoothly transitions through values between -1 and 1 as \( \theta \) varies.

The sine function also plays an essential role in the unit circle, where each point \((\cos \theta, \sin \theta)\) on the circle corresponds to the coordinates of the angle \( \theta \). This connection helps to visualize how sine relates to an angle and its respective triangle's dimensions.

Exploring the Cosine Function

Like the sine function, the cosine function is a significant trigonometric identity. It is expressed as \( \cos \theta \) and represents the ratio of the adjacent side to the hypotenuse in a right triangular context. The cosine function, like sine, is periodic and repeats every \( 2\pi \) radians or 360 degrees.

• The cosine of 0 degrees or 0 radians is 1.
• The cosine of 90 degrees or \( \frac{\pi}{2} \) radians is 0.
• It also smoothly oscillates between -1 and 1.

In the context of the unit circle, \( \cos \theta \) corresponds to the x-coordinate of the point at an angle \( \theta \). This relationship explains its significance in defining the secant function. The secant function is the reciprocal of cosine, mathematically expressed as \( \sec \theta = \frac{1}{\cos \theta} \). Understanding this reciprocal helps simplify various trigonometric expressions.

The Tangent Function and its Simplification

The tangent function \( \tan \theta \) is another core trigonometric identity, derived from the ratio of sine to cosine, specifically \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This function reflects the ratio of the opposite side to the adjacent side in a right triangle.Some key properties of the tangent function include:

• It is undefined where \( \cos \theta = 0 \), leading to vertical asymptotes on its graph, such as at \( \theta = \frac{\pi}{2} \) and \( \theta = \frac{3\pi}{2} \).
• Tangent is periodic with a cycle of \( \pi \) radians or 180 degrees because only a half cycle of sine and cosine are needed to repeat the ratio.
• It grows rapidly between its asymptotes, transitioning smoothly through zero.

In the context of our exercise, simplifying \( \sin \theta \sec \theta \) involved recognizing that this expression simplifies to \( \tan \theta \). This step demonstrates the importance of expressing trigonometric functions in simpler terms using their identities.