Problem 108
Question
Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\frac{\cos 3 \beta}{\cos \beta}=1-4 \sin ^{2} \beta$$
Step-by-Step Solution
Verified Answer
The identity is indeed true as demonstrated in the step-by-step solution. Verification through graphical method also confirms this.
1Step 1: Replace Complex Trigonometric Function
Replace \(\cos 3\beta \) using the triple angle identity \( \cos 3\beta = 4\cos^{3}\beta - 3\cos\beta\). So the equation to prove becomes \(\frac{4\cos^3\beta - 3\cos\beta}{\cos\beta} = 1 - 4\sin^2\beta \)
2Step 2: Simplify the Left Side
Simplify by dividing each term in the numerator by \(\cos\beta\). This gives us \(4\cos^2\beta - 3 = 1 - 4\sin^2\beta \)
3Step 3: Use Pythagorean Identity
Replace \(\cos^2\beta\) with \(1-\sin^2\beta\). This gives us \(4(1-\sin^2\beta) - 3 = 1 - 4\sin^2\beta\)
4Step 4: Simplify Both Sides
Simplify both sides of the equation which gives us \(4 - 4\sin^2\beta - 3 = 1 - 4\sin^2\beta\). Thus the identity is verified.
5Step 5: Use a Graphing Utility
You should plot both sides of the equation \(\frac{\cos 3\beta}{\cos \beta}\) and \(1 - 4\sin^2\beta\) on the same set of axes to visually confirm they are the same. Both the plots coincide which verifies the given identity graphically.
Key Concepts
Triple Angle IdentitiesPythagorean Identity in TrigonometrySimplifying Trigonometric ExpressionsGraphical Representation of Trigonometric Functions
Triple Angle Identities
Triple angle identities in trigonometry are an extension of the double angle identities and give us expressions for the sine, cosine, and tangent of triple angles in terms of single angles. For instance, the triple angle identity for cosine is expressed as \(\cos 3\theta = 4\cos^3\theta - 3\cos\theta\). This can be particularly useful when dealing with more complex trigonometric equations or when simplifying expressions.
These identities are not just pulled out of thin air; they are derived from the standard angle sum identities and can be proven through substitution and algebraic manipulation. In practice, these identities can simplify seemingly complex problems, such as turning a trigonometric function of a triple angle into a function of a single angle, making the problem more manageable.
These identities are not just pulled out of thin air; they are derived from the standard angle sum identities and can be proven through substitution and algebraic manipulation. In practice, these identities can simplify seemingly complex problems, such as turning a trigonometric function of a triple angle into a function of a single angle, making the problem more manageable.
Pythagorean Identity in Trigonometry
The Pythagorean identity is a cornerstone of trigonometry, based on the Pythagorean theorem related to right triangles. It states that for any angle \(\theta\), the square of the sine plus the square of the cosine of that angle equals 1, written mathematically as \(\sin^2\theta + \cos^2\theta = 1\).
This fundamental relationship allows us to interchange \(\sin^2\theta\) and \(\cos^2\theta\) with \(1 - \cos^2\theta\) and \(1 - \sin^2\theta\) respectively. It is an essential tool for simplifying trigonometric expressions and for proving the trigonometric identities, like the one presented in our example exercise. By applying the Pythagorean identity, complex identities can often be reduced to more familiar algebraic forms.
This fundamental relationship allows us to interchange \(\sin^2\theta\) and \(\cos^2\theta\) with \(1 - \cos^2\theta\) and \(1 - \sin^2\theta\) respectively. It is an essential tool for simplifying trigonometric expressions and for proving the trigonometric identities, like the one presented in our example exercise. By applying the Pythagorean identity, complex identities can often be reduced to more familiar algebraic forms.
Simplifying Trigonometric Expressions
Simplifying trigonometric expressions is crucial to solving trigonometry problems efficiently. This process includes a variety of techniques such as using fundamental identities like Pythagorean and triple angle identities, factoring, and combining like terms.
To simplify effectively, recognizing patterns and understanding the properties of trigonometric functions is key. For example, seeing a \(\cos^3\beta\) may prompt one to think of the triple angle identity, where this expression could be part of a larger identity transformation. Algebraic manipulations such as canceling terms or factoring are common methods used in conjunction with trigonometric identities to simplify an expression to its most reduced form.
To simplify effectively, recognizing patterns and understanding the properties of trigonometric functions is key. For example, seeing a \(\cos^3\beta\) may prompt one to think of the triple angle identity, where this expression could be part of a larger identity transformation. Algebraic manipulations such as canceling terms or factoring are common methods used in conjunction with trigonometric identities to simplify an expression to its most reduced form.
Graphical Representation of Trigonometric Functions
Graphical representation of trigonometric functions involves plotting sine, cosine, tangent, and other trigonometric functions on a set of axes (usually Cartesian coordinates). By visualizing these functions graphically, we can gain a deeper understanding of their behavior and properties, such as periodicity and amplitude.
In the context of verifying trigonometric identities, graphing both sides of the identity on the same graph can serve as both a visual confirmation and a troubleshooting tool. If the graphs overlap, it implies that the two expressions are equivalent for all values in the domain. This graphical approach is particularly helpful as it drives home the concept that identities hold true for all values of the variable, not just a specific case.
In the context of verifying trigonometric identities, graphing both sides of the identity on the same graph can serve as both a visual confirmation and a troubleshooting tool. If the graphs overlap, it implies that the two expressions are equivalent for all values in the domain. This graphical approach is particularly helpful as it drives home the concept that identities hold true for all values of the variable, not just a specific case.
Other exercises in this chapter
Problem 107
Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\frac{\sin 3 y}{\sin y}=1-2 \sin ^{2} y+2 \cos ^{2} y$$
View solution Problem 107
The area of a rectangle inscribed in one arc of the graph of \(y=\cos x\) (see figure) is given by \(A=2 x \cos x, \quad 0 \leq x \leq \frac{\pi}{2}\). (a) Use
View solution Problem 108
Determine whether the statement is true or false. Justify your answer. $$\sin \theta \csc \theta=1$$
View solution Problem 109
Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\csc \frac{u}{2}=\pm \sqrt{\frac{2 \csc u}{\csc u-\cot u}}$$
View solution