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Chapter 5: Problem 9

Simplify each of the following trigonometric expressions. $$ \frac{\csc x}{\cot x} $$

Short Answer

Expert verified
The simplified expression is \( \sec x \).

Step by step solution

01

Express Functions in Terms of Sine and Cosine

Recall the definitions of cosecant and cotangent in terms of sine and cosine. \[ \csc x = \frac{1}{\sin x} \quad \text{and} \quad \cot x = \frac{\cos x}{\sin x} \] Therefore, the expression \( \frac{\csc x}{\cot x} \) becomes:\[ \frac{\frac{1}{\sin x}}{\frac{\cos x}{\sin x}} \]
02

Simplify the Fraction Division

When dividing fractions, multiply by the reciprocal of the divisor. Thus:\[ \frac{\frac{1}{\sin x}}{\frac{\cos x}{\sin x}} = \frac{1}{\sin x} \times \frac{\sin x}{\cos x} \]
03

Cancel Common Factors

Cancel out \( \sin x \) in the numerator and denominator:\[ \frac{1}{\cos x} \]This results in the expression for secant: \[ \sec x = \frac{1}{\cos x} \]

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Key Concepts

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

Cosecant
Cosecant is one of the six fundamental trigonometric functions, and it's represented as \( \csc \). It is the reciprocal of the sine function. This means that \( \csc x \) is equal to \( \frac{1}{\sin x} \).
It is particularly useful in situations where direct values of the sine function are available, as it allows for simple conversions:
  • If \( \sin x = 0.5 \), then \( \csc x = \frac{1}{0.5} = 2 \).
  • Cosecant is undefined when sine is zero, as division by zero is not allowed.
Understanding cosecant is crucial in solving trigonometric equations and simplifying expressions involving trigonometric identities. In general, it allows for expressing more complex trig functions in a different yet simplified way.
Cotangent
Cotangent, denoted as \( \cot \), is another important trigonometric function. It is defined as the reciprocal of the tangent function, expressed by \( \cot x = \frac{\cos x}{\sin x} \). Here, cosine is the adjacent over hypotenuse ratio, while sine is the opposite over hypotenuse ratio.
Its properties and relationships with other trig functions make it a powerful tool for simplifying expressions:
  • When \( \tan x = 3 \), then \( \cot x = \frac{1}{3} \).
  • Cotangent is undefined at points where sine equals zero, because you can't divide by zero.
Cotangent plays an integral role in trigonometric identities and equations. It helps move between different representations of angles and their relationships within a triangle.
Sine and Cosine
Sine and cosine are perhaps the most fundamental of all trigonometric functions. They describe relationships in right-angled triangles and are periodic with a cycle of \( 2\pi \).
  • Sine, \( \sin x \), is defined as the ratio of the length of the opposite side to the length of the hypotenuse.
  • Cosine, \( \cos x \), is the ratio of the length of the adjacent side to the hypotenuse.
These functions are the building blocks for other trigonometric functions, including tangent, secant, cosecant, and cotangent. Through various identities, they allow for the transformation and simplification of trigonometric equations.
Identities Involving Sine and Cosine
Exploring identities can greatly aid in the simplification of expressions like the one found in the original exercise. For instance:
  • \( \sin^2 x + \cos^2 x = 1 \)
  • \( \tan x = \frac{\sin x}{\cos x} \)
Mastering sine and cosine provides a solid foundation for comprehending more complex trigonometric concepts and solving related mathematics problems efficiently.

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Most popular questions from this chapter

Find the exact value of \(\sin 15^{\circ}\) in two ways, using sum and difference identities and half-angle identities; then show that they are equal.

Consider the triangle below, where the vertex angle measures \(\theta\), the equal sides measure \(a\), the height is \(h\), and half the base is \(b\). (In an isosceles triangle, the perpendicular dropped from the vertex angle divides the triangle into two congruent triangles.) The two triangles formed are right triangles. In the right triangles, \(\sin \left(\frac{\theta}{2}\right)=\frac{b}{a}\) and \(\cos \left(\frac{\theta}{2}\right)=\frac{h}{a}\). Multiply each side of each equation by \(a\) to get \(b=a \sin \left(\frac{\theta}{2}\right), h=a \cos \left(\frac{\theta}{2}\right)\). The area of the entire isosceles triangle is \(A=\frac{1}{2}(2 b) h=b h\). Substitute the values for \(b\) and \(h\) into the area formula. Show that the area is equivalent to \(\frac{a^{2}}{2} \sin \theta\).

Computer sales are generally subject to seasonal fluctuations. An analysis of the sales of a computer manufacturer during 2008-2010 is approximated by the function $$ s(t)=0.098 \cos ^{2} t+0.387 \quad 1 \leq t \leq 12 $$ where \(t\) represents time in quarters ( \(t=1\) represents the end of the first quarter of 2008\()\), and \(s(t)\) represents computer sales (quarterly revenue) in millions of dollars. Use a double-angle identity to express \(s(t)\) in terms of the cosine function.

Use the half-angle identities to find the exact values of the trigonometric expressions. $$ \tan \left(67.5^{\circ}\right) $$

Verify the identities. $$ \sec \left(\frac{A}{2}\right)=\pm|\csc A| \sqrt{2-2 \cos A} $$
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