Problem 41 - Precalculus: A Concise Course

Question

Use trigonometric identities to transform the left side of the equation into the right side \((0<\theta<\pi / 2)\). $$ (1+\sin \theta)(1-\sin \theta)=\cos ^{2} \theta $$

Step-by-Step Solution

Answer

Yes, indeed by expanding the left side of the equation and using the Pythagorean identity we can show that \( (1+sin \theta)(1-sin \theta) = \cos ^{2} \theta \).

Step 1: Expand the left-hand side

Expand the left-hand side of the equation using the formula for the product of two binomials \( (1+sin \theta)(1-sin \theta) = 1 - sin^{2} \theta \)

Step 2: Apply the Pythagorean identity

Then, replace \( sin^{2} \theta \) with \( 1 - cos^{2} \theta \) from the Pythagorean identity. This is valid because \( sin^{2} \theta + cos^{2} \theta = 1 \), so you can rearrange it to find \( sin^{2} \theta = 1 - cos^{2} \theta \). Substituting it into the equation gives \( 1 - (1 - cos^{2} \theta) = cos^{2} \theta \)

Step 3: Simplify the equation

Finally, simplify the equation. \( 1 - (1 - cos^{2} \theta) = cos^{2} \theta \) simplifies to just \( cos^{2} \theta = cos^{2} \theta \), proving that the left side of the equation is equal to the right side.

Key Concepts

Pythagorean Identity

The Pythagorean identity is a cornerstone of trigonometry, providing an essential relationship between the sine and cosine of an angle. This identity states that for any angle \( \theta \), the square of the sine of \( \theta \) plus the square of the cosine of \( \theta \) equals one: \[ sin^{2} \theta + cos^{2} \theta = 1 \. \]
Imagine you have a right triangle with sides \( a \), \( b \), and hypotenuse \( c \). The Pythagorean theorem tells us that \( a^{2} + b^{2} = c^{2} \). When we deal with the unit circle (where the radius \( c = 1 \)), the legs of the triangle can be viewed as the sine and cosine of the angle \( \theta \) with respect to the unit circle: \( sin \theta = \frac{a}{c} \) and \( cos \theta = \frac{b}{c} \). Substituting these into the Pythagorean theorem gives us the Pythagorean identity.
It is extremely useful in transforming trigonometric expressions, as seen in the original exercise. Knowing this identity allows the alteration of trigonometric functions within an equation to facilitate simplification, as demonstrated in the expansion and substitution steps in the provided solution.

Binomial Multiplication

Binomial multiplication is a fundamental skill in algebra, referring to the process of expanding the product of two binomials. A binomial is a polynomial with two terms, usually written in the form \( (a + b) \). When multiplying two binomials, such as \( (1 + sin \theta)(1 - sin \theta) \), we apply the distributive property — which is often referred to as FOIL (First, Outer, Inner, Last) when dealing with binomials.

• First: Multiply the first terms in each binomial.
• Outer: Multiply the outer terms.
• Inner: Multiply the inner terms.
• Last: Multiply the last terms in each binomial.

This process results in four terms which are then combined to obtain the final expanded form. In our exercise, the expansion simplifies because the middle terms cancel out, leaving us with \( 1 - sin^{2} \theta \).
Through practice, binomial multiplication becomes an automatic process that significantly aids in solving more complex algebraic and trigonometric equations.

Trigonometric Equation

A trigonometric equation is an equation that involves trigonometric functions of an unknown angle \( \theta \). These equations are quite prevalent in various areas of mathematics and physics, as they can describe waves, oscillations, and circular motion among other phenomena.
Solving such equations usually involves a few typical steps. First, one should manipulate the equation to isolate the trigonometric function. This might include using algebraic techniques like the binomial multiplication seen in the exercise or applying trigonometric identities like the Pythagorean identity.
Once simplified, the next step may involve inverse trigonometric functions to solve for the angle \( \theta \), or further manipulations and identities that transform the equation into a familiar form.

Key to Success

• Familiarity with basic trigonometric identities.
• Proficiency in algebraic manipulations.
• Strategic use of inverse trigonometric functions where necessary.

By mastering these foundational concepts, one can tackle a wide range of trigonometric equations with confidence.