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Chapter 6: Problem 40

In Exercises \(27-80,\) verify the given identities. $$\cos ^{2} x \csc x=\csc x-\sin x$$

Short Answer

Expert verified
After following the steps to simplify the equation and using trigonometric identities, we can see the equation holds true for all valid values of \(x\). Thus, the given trigonometric identity \(\cos ^{2} x \csc x=\csc x-\sin x\) is verified.

Step by step solution

01

Remember definitions

Remember the definition of the trigonometric functions involved. In this case: \(\cos^2 x\) is the square of the cosine function, \(\csc x\) is the reciprocal of the sine function, i.e., \(1/\sin x\), and the \(\sin x\) is just the sine function.
02

Divide both sides by \(\csc x\)

Dividing both sides by \(\csc x\) simplifies the equation. This gives: \(\cos ^{2} x = 1 - \sin x\). This is done because \(\csc x \times\sin x = 1\) (by definition, since \(\csc x\) is \(1/\sin x\)).
03

Convert \(\cos ^{2} x\) to \(1 - \sin ^{2} x\)

Now, use the identity \(\sin^2 x + \cos^2 x = 1\), which can be rearranged to \(\cos ^{2} x = 1 - \sin ^{2} x\).
04

Substitute

Substitute \(\cos ^{2} x\) in the equation from Step 3 into the equation from Step 2, which results in: \(1 - \sin ^{2} x = 1 - \sin x\).
05

Simplify

Squaring \(\sin x\) and subtracting it from 1 results in \(0\), which confirms that the original equation is indeed correct. This gives \(0 = 0\), verifying the identity.

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Key Concepts

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

Trigonometric Functions
Trigonometric functions are fundamental in precalculus and calculus. They are used to relate angles and sides in right triangles, and they appear in various mathematical fields, including geometry and physics.
Understanding their definitions is key to solving many problems.
Trigonometric functions include:
  • Sine (\( \sin x \)): This function represents the ratio of the opposite side to the hypotenuse in a right triangle.
  • Cosine (\( \cos x \)): This function denotes the ratio of the adjacent side to the hypotenuse.
  • Cosecant (\( \csc x \)): The reciprocal of the sine function, \( \csc x = 1/\sin x \).
These functions help solve various equations by allowing conversion between different forms and simplification of expressions.
Trigonometric Identities Verification
Verifying trigonometric identities is a crucial skill in precalculus exercises.
It involves proving that two different expressions, possibly involving trigonometric functions, are equivalent. This requires familiarity with trigonometric identities and operations.A common strategy is:
  • Understanding the expressions: Break down both sides using known identities and definitions for trigonometric functions.
  • Applying algebraic manipulations: Use operations like factoring, expanding, and simplifying expressions.
  • Using standard identities: These include Pythagorean identities such as \( \sin^2 x + \cos^2 x = 1 \).
In this process, it's often helpful to work on one side of the equation to transform it into the other side, verifying their equivalency.
Precalculus Exercises
Precalculus exercises provide an excellent foundation for understanding concepts that will be further explored in calculus.
These exercises often involve the use of algebraic manipulation and trigonometric identities to solve complex equations. Successful tackling of these problems includes:
  • Identifying Key Concepts: Understanding what is being asked and what principles or identities may be applicable.
  • Strategic Problem Solving: Breaking the problem into smaller, more manageable parts.
  • Verification Steps: Checking each step of your work against known identities to ensure accuracy.
Precalculus is not just about finding the solution, but also about understanding the reasoning and methodology behind each step, paving the way for more advanced studies in mathematics.

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Most popular questions from this chapter

In Exercises \(69-82,\) prove the given identities. $$\tan (\pi+x)=\tan x$$

Write the given expressions as a product of two trigonometric functions. $$\cos 3 x-\cos 5 x$$

Verify the given identities. $$\sin 4 x=4 \sin x \cos x\left(1-2 \sin ^{2} x\right)$$

The horizontal range of a projectile fired with an initial velocity of 40 meters per second at an angle \(\theta\) is given by \(R=\frac{40^{2} \sin 2 \theta}{9.8} .\) Find \(R\) to four decimal places if it is known that \(\sin \theta=0.3\) and \(\theta\) is in the first quadrant.

Derive the following sum-to-product identity. $$\sin a-\sin b=2 \cos \left(\frac{a+b}{2}\right) \sin \left(\frac{a-b}{2}\right)$$
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