Problem 63 - Algebra 2

Question

Which equation is NOT true? \(\begin{array}{ll}{\mathbf{F} \cos ^{2} \theta=1-\sin ^{2} \theta} & {\text { G. } \cot ^{2} \theta=\csc ^{2} \theta-1} \\ {\text { H. } \sin ^{2} \theta=\cos ^{2} \theta-1} & {\text { I. } \tan ^{2} \theta=\sec ^{2} \theta-1}\end{array}\)

Step-by-Step Solution

Answer

H. \( \sin ^{2} \theta = \cos ^{2} \theta - 1\) is not true.

Step 1: Check the First Equation

The first equation is \(F \cos ^{2} \theta = 1 - \sin ^{2} \theta\). This can be rewritten as \(\cos ^{2} \theta = 1 - \sin ^{2} \theta\). This is a Pythagorean identity and thus is true.

Step 2: Check the Second Equation

The second equation is \( \cot ^{2} \theta = \csc ^{2} \theta - 1\). This is another example of the Pythagorean identity and is thus also true.

Step 3: Check the Third Equation

The third equation is \(\sin ^{2} \theta = \cos ^{2} \theta -1\). By rearranging terms, we get \(\sin ^{2} \theta + 1 = \cos ^{2} \theta\). This does not match the standard Pythagorean identity. Therefore, this equation is not true.

Step 4: Check the Fourth Equation

The fourth equation is \(\tan ^{2} \theta = \sec ^{2} \theta - 1\). This is one of the Pythagorean identities as well, meaning this equation is true.

Key Concepts

Pythagorean Identities

Pythagorean Identities are one of the fundamental types of identities in trigonometry. They are derived from the Pythagorean Theorem in geometry, which is expressed as: \[ a^2 + b^2 = c^2 \] when applied to a right triangle. In trigonometry, these identities connect the squares of the sine and cosine functions:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)
• \( 1 + \tan^2 \theta = \sec^2 \theta \)
• \( 1 + \cot^2 \theta = \csc^2 \theta \)

These identities can be incredibly useful in simplifying expressions and solving trigonometric equations. In practice, recognizing the Pythagorean Identities allows students to rewrite expressions in different forms, which is crucial in solving many trigonometric problems. If you're facing an equation like \( \cos^2 \theta = 1 - \sin^2 \theta \), it's a direct application of the first Pythagorean identity, which confirms its validity without further calculations.

Trigonometric Equations

Trigonometric Equations are equations that involve trigonometric functions and are solved over specific intervals or throughout an infinite domain. Understanding these equations often requires both knowledge of trigonometric identities, such as the Pythagorean Identities, and algebraic manipulation.

To solve this type of equation, it's necessary to apply well-known identities that simplify or rearrange the terms. Let's say we encounter an equation like \( \sin^2 \theta = \cos^2 \theta - 1 \). First, apply a Pythagorean Identity to both sides or convert the equation into a standard form that lets you evaluate the trigonometric function in question.

• Identify known identities that match terms on either side of the equation.
• Substitute these identities to simplify the equation.
• Isolate the trigonometric function to solve for \( \theta \).
• Check the solutions within the required domain or interval.

Misunderstandings can arise, especially when signs or terms are transferred improperly. That's why verifying through substitution often reveals unruly equations, such as when terms don't neatly match known identities, indicating that they may be incorrect.

Trigonometric Functions

Trigonometric Functions, such as sine, cosine, tangent, cotangent, secant, and cosecant, represent the ratios of different sides of a right triangle. They are foundational in describing the relationships between angles and distances, not just limited to geometry but in various fields of science and engineering.

Each function has its own definition based on the angle \( \theta \):

• \( \sin \theta = \frac{opposite}{hypotenuse} \)
• \( \cos \theta = \frac{adjacent}{hypotenuse} \)
• \( \tan \theta = \frac{opposite}{adjacent} \)
• \( \cot \theta = \frac{adjacent}{opposite} \)
• \( \sec \theta = \frac{hypotenuse}{adjacent} \)
• \( \csc \theta = \frac{hypotenuse}{opposite} \)

These functions also relate closely through various identities and transformations. For example, \( \tan \theta \) relates to \( \sec \theta \) through the identity \( \tan^2 \theta + 1 = \sec^2 \theta \). Understanding how these relationships work is key to solving trigonometric equations and understanding graphs of these functions, which oscillate based on the input angle. Practice with these functions helps to foresee outcomes, like verifying if particular trigonometric equations are plausible.