Problem 6 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

\( \tan \theta \csc \theta \) simplifies to \( \sec \theta. \)

Step 1: Express Tangent in Terms of Sine and Cosine

The tangent of an angle \( \theta \) can be expressed using sine and cosine as follows:\[ \tan \theta = \frac{\sin \theta}{\cos \theta}. \]

Step 2: Express Cosecant in Terms of Sine

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

Step 3: Multiply the Expressions

Substitute the expressions from Steps 1 and 2 into the original trigonometric expression:\[ \tan \theta \csc \theta = \left( \frac{\sin \theta}{\cos \theta} \right) \left( \frac{1}{\sin \theta} \right). \]

Step 4: Simplify the Expression

Multiply the fractions obtained in Step 3:\[ \tan \theta \csc \theta = \frac{\sin \theta}{\cos \theta} \times \frac{1}{\sin \theta} = \frac{\sin \theta}{\sin \theta \cos \theta}. \]This simplifies to:\[ \frac{1}{\cos \theta} \] because \( \sin \theta \) in the numerator and denominator cancels out.

Step 5: Recognize the Simplified Trigonometric Function

The simplified expression \( \frac{1}{\cos \theta} \) is the definition of secant:\[ \sec \theta. \]

Key Concepts

Tangent

Tangent is one of the fundamental trigonometric functions. It relates the lengths of the opposite side to the adjacent side in a right-angled triangle.

• The tangent of an angle \( \theta \) is defined as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
• This means that tangent is the ratio of sine to cosine, or the ratio of the opposite side to the adjacent side in a right triangle.
• Tangent can also be thought of as the slope of a line from the origin to a point on the unit circle.

Using this relationship helps simplify expressions when combined with other trigonometric functions. Replace anything involving tangent with sine and cosine to simplify problems.

Cosecant

Cosecant is another trigonometric function. It is closely related to the sine function as it is its reciprocal.

• The cosecant of an angle \( \theta \) is defined as \( \csc \theta = \frac{1}{\sin \theta} \).
• This means that cosecant is the reciprocal of sine, which is the inverse relationship between the hypotenuse and the opposite side.
• In the unit circle, it is the reciprocal of the vertical coordinate of the point lying on the terminal side of the angle.

Understanding how cosecant works with sine helps to interchange and simplify complex trigonometric identities.

Secant

Secant is a trigonometric function that serves as the reciprocal of the cosine function.

• The secant of an angle \( \theta \) is \( \sec \theta = \frac{1}{\cos \theta} \).
• In a right triangle, secant represents the ratio of the hypotenuse to the adjacent side.
• Secant extends beyond a 90-degree angle and can provide insights into the growth of functions in trigonometric calculus.

In trigonometric equations, transforming terms into secant by using its relation to cosine can streamline solving processes, as seen with the simplification of expressions like \( \tan \theta \csc \theta \).

Sine and Cosine

Sine and cosine are the core building blocks of trigonometry. They measure projections in a right triangle.

• Sine of an angle \( \theta \), denoted by \( \sin \theta \), is defined as the ratio of the length of the opposite side to the hypotenuse.
• Cosine of an angle \( \theta \), or \( \cos \theta \), is the ratio of the adjacent side to the hypotenuse.
• They are related by the identity \( \sin^2 \theta + \cos^2 \theta = 1 \).

Sine and cosine allow the transformation of more complex trigonometric expressions into simpler ratio forms. Simplifying expressions with these forms can help solve problems more efficiently, as demonstrated in the simplification process where tangent and cosecant are both expressed in terms of sine and cosine.