Problem 3 - Algebra 2

Question

Verify each identity. $$ \cos \theta \tan \theta=\sin \theta $$

Step-by-Step Solution

Answer

The identity has been verified as the simplified expression \(\sin \theta = \sin \theta\) is an identity itself. Hence, the original identity \(\cos \theta \tan \theta = \sin \theta\) holds true for all values of \(\theta\) where both sides of the equation are defined.

Step 1: Expression for Tangent

Rewrite the tangent function as a quotient of sine over cosine: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)

Step 2: Substitute the expression for Tangent into the original equation

Now replace \(\tan \theta\) in the original equation by its equivalent \(\frac{\sin \theta}{\cos \theta}\). So, the equation \(\cos \theta\tan \theta = \sin \theta\) transforms into: \(\cos \theta \cdot \frac{\sin \theta}{\cos \theta} = \sin \theta\)

Step 3: Simplify

Simplify the expression by cancelling the \(\cos \theta\) on the left side of the equation which leaves us with:\( \sin \theta = \sin \theta \)

Key Concepts

Sine and Cosine Functions

Sine and cosine are fundamental trigonometric functions that express relationships between the sides and angles of a right triangle.
They help in describing the oscillations of waves, circles, and various natural phenomena.

• Sine Function (\(\sin\theta\)): It represents the ratio of the opposite side to the hypotenuse in a right triangle. Its values vary from -1 to 1.
• Cosine Function (\(\cos\theta\)): It represents the ratio of the adjacent side to the hypotenuse. Like sine, cosine values also range from -1 to 1.

These functions are periodic, repeating their values in regular intervals. For example, both sine and cosine functions have a period of 360 degrees or \(2\pi\) radians.
Understanding these functions is essential in various fields such as physics, engineering, and everyday calculations.

Tangent as a Quotient

The tangent function is one of the primary trigonometric functions and is especially important because it connects sine and cosine.
It is defined as the ratio of the sine function to the cosine function.The equation \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) reflects this relationship precisely.
This definition emphasizes how the tangent function grows rapidly as the cosine of the angle approaches zero.

• The tangent function is undefined where the cosine is zero because division by zero is not possible.
• Tangent repeats every 180 degrees or \(\pi\) radians, making it a periodic function with important applications such as in waves and signal processing.

Understanding tangent as a quotient simplifies many trigonometric calculations and helps in visualizing angles and rotations.

Verifying Identities

Verifying trigonometric identities is a crucial skill in mathematics which involves proving that one side of an equation is equivalent to the other.
This process strengthens problem-solving skills and aptitude for mathematical logic.To verify identities, consider the following simple steps:

• Identify known trigonometric identities and how they relate to the expressions involved.
• Simplify expressions on both sides of the equation, using algebraic manipulations.
• Replace trigonometric functions with their known identities, if necessary, to simplify the comparison.
• Check if both sides of the equation eventually simplify to the same expression, thus verifying the identity.

In the example exercise, after substituting the expression of tangent as \(\frac{\sin \theta}{\cos \theta}\) and cancelling similar terms, both sides equate to \(\sin \theta\), confirming the equation is true.
Such exercises help to appreciate the interrelationships between trigonometric functions and their properties.