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Chapter 6: Problem 9

In Exercises \(9-14\), show that the given equations are not identities. $$\cos x=0.5$$

Short Answer

Expert verified
The equation \( \cos x = 0.5 \) is not an identity because it doesn't hold true for all permissible values of \( x \). Example, for \( x=90掳 \), we get \( \cos 90掳 = 0 \), and not 0.5. Thus, proving our assertion.

Step by step solution

01

Understanding the definition of an Identity

First, it's vital to recognize what exactly a trigonometric identity is. An identity is an equation that is valid for all values of the variable. In trigonometry, identities often involve trigonometric functions such as sine, cosine, tangent, etc. In this case, the equation is \( \cos x=0.5 \), where \( x \) is the variable.
02

Check the given equation

The next step is to check the equation to verify if it is an identity or not. The equation \( \cos x=0.5 \) suggests that the cosine of the angle \( x \) always equals to 0.5. This is the step where we check if this equation holds for any value of \( x \) or not.
03

Substituting values

We can now substitute different values of \( x \) in the equation to see if \( \cos x \) equals 0.5 every time. For instance, if we insert \( x=90掳 \) in the equation we get \( \cos 90掳 \), which is 0 not 0.5.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosine Function
The cosine function is one of the fundamental functions in trigonometry. It is usually written as \( \cos x \), where \( x \) is the angle measured in degrees or radians. The cosine function is periodic, with a period of \( 360^{\circ} \) or \( 2\pi \) radians. This means its pattern repeats every full circle.
  • The range of the cosine function is from \(-1\) to \(1\). This indicates that no matter the angle \( x \), \( \cos x \) can never be less than \(-1\) or greater than \(1\).
  • The cosine of some common angles: \( \cos 0^{\circ} = 1 \), \( \cos 90^{\circ} = 0 \), \( \cos 180^{\circ} = -1 \), and \( \cos 360^{\circ} = 1 \).
Understanding these cosine values helps us evaluate various trigonometric problems. For a specific equation like \( \cos x = 0.5 \), knowing the behavior of cosine is crucial.
Equations
In mathematics, an equation is a statement that asserts the equality of two expressions. Trigonometric equations involve trigonometric functions like sine, cosine, or tangent.
  • A specific trigonometric equation like \( \cos x = 0.5 \) asserts that for some angle \( x \), the cosine value is \(0.5\).
  • It does not automatically mean that this condition is valid for all angles \( x \). This distinction is crucial in determining if an equation is an identity.
To verify the given equation, we'll test it using different angles. If there are only certain angles where \( \cos x = 0.5 \), then it is not an identity but a conditional equation.
Substitution Method
The substitution method involves replacing a variable with a specific value to test if an equation holds true for that instance. This is useful for verifying if a trigonometric equation is an identity.
  • In our example, substituting \( x = 60^{\circ} \) into \( \cos x = 0.5 \) works because \( \cos 60^{\circ} = 0.5 \).
  • However, substituting \( x = 90^{\circ} \) gives \( \cos 90^{\circ} = 0 \), not \( 0.5 \). Thus, \( \cos x = 0.5 \) is not true for every \( x \).
By substituting and checking various angles, we confirm that \( \cos x = 0.5 \) is not a universal truth for all \( x \). It applies only to specific angles, proving it's not an identity.

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Most popular questions from this chapter

In Exercises \(69-82,\) prove the given identities. $$\cos (x-y) \cos (x+y)=\cos ^{2} x \cos ^{2} y-\sin ^{2} x \sin ^{2} y$$

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