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Chapter 5: Problem 1

Fill in the blank to complete the trigonometric identity. \(\frac{\sin u}{\cos u}=\)________

Short Answer

Expert verified
The blank can be filled with tan(u).

Step by step solution

01

Understand the Trigonometric Identity

The given expression is the quotient of sin(u) and cos(u). In trigonometry, there exists an identity which relates these two functions, also known as a quotient identity. It is the tangent function denoted as tan(u).
02

Formulate the Identity

Based on the understanding of trigonometric identities, specifically, the formula for tan(u) which states that tan(u) = sin(u)/cos(u), so the blank can be filled with tan(u) as this is the ratio of sin(u) to cos(u).

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Key Concepts

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

Sine Function
The sine function, often denoted as \( \sin \theta \), is a fundamental component of trigonometry. This function relates to the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle.
In the unit circle, which is a circle with a radius of 1 centered at the origin of a coordinate plane, sine represents the y-coordinate of the point where the terminal side of an angle intersects the circle.
  • When the angle \( \theta \) is 0 degrees, \( \sin \theta = 0 \).
  • At 90 degrees, \( \sin \theta = 1 \).
  • The sine function is periodic, repeating every 360 degrees or \( 2\pi \) radians.
Understanding the sine function is crucial for solving various trigonometric identities, as it often forms part of expressions involving other trigonometric functions.
Cosine Function
The cosine function, written as \( \cos \theta \), complements the sine function in trigonometry. It describes the ratio of the adjacent side to the hypotenuse in a right triangle. Like the sine function, the cosine function can also be visualized on the unit circle. Here, it signifies the x-coordinate of the point where the terminal side of the angle intersects the circle.
  • At 0 degrees, \( \cos \theta = 1 \).
  • At 90 degrees, \( \cos \theta = 0 \).
  • Cosine is also a periodic function with a period of 360 degrees or \( 2\pi \) radians.
The cosine and sine functions are closely related and are often used together in trigonometric identities like the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \). This interplay is crucial in understanding and using identities in trigonometry.
Tangent Function
The tangent function, represented as \( \tan \theta \), is a fundamental trigonometric function that often appears in various identities and equations. It is defined as the ratio of the sine and cosine functions: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This identity is crucial for simplifying expressions and solving problems in trigonometry.
  • The tangent of an angle in a right triangle can also be expressed as the ratio of the opposite side to the adjacent side.
  • The tangent function exhibits periodicity, repeating every 180 degrees or \( \pi \) radians.
  • Tangent can be undefined when the cosine function is zero, such as at 90 degrees or 270 degrees, because you cannot divide by zero.
The understanding of how tangent relates to sine and cosine is pivotal when working with more complex trigonometric identities and expressions. This relationship simplifies many trigonometric problems and is an essential concept in trigonometry.

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Most popular questions from this chapter

Reducing Powers, use the power-reducing formulas to rewrite the expression in terms of the first power of the cosine. $$\sin ^{2} 2 x \cos ^{2} 2 x$$

Using Half-Angle Formulas, use the half-angle formulas to simplify the expression. $$-\sqrt{\frac{1-\cos 8 x}{1+\cos 8 x}}$$

Using Half-Angle Formulas, (a) determine the quadrant in which \(u\) 2 lies, and (b) find the exact values of \(\sin (u\) 2), \(\cos (u\) 2), and \(\tan (u\) 2) using the half-angle formulas. $$\cos u=7 / 25, \quad 0

Using Sum-to-Product Formulas, use the sum-to-product formulas to find the exact value of the expression. $$ \sin 75^{\circ}+\sin 15^{\circ} $$

Reducing Powers, use the power-reducing formulas to rewrite the expression in terms of the first power of the cosine. $$\tan ^{2} 2 x \cos ^{4} 2 x$$
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