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Chapter 6: Problem 35

Verify the identity. $$\tan 3 u=\frac{\tan u\left(3-\tan ^{2} u\right)}{1-3 \tan ^{2} u}$$

Short Answer

Expert verified
The identity is verified because both sides simplify to the same expression.

Step by step solution

01

Recall the triple angle identity for tangent

The triple angle identity for tangent is given by \( \tan(3u) = \frac{3\tan(u) - \tan^3(u)}{1 - 3\tan^2(u)} \). This is the identity we will use to verify the problem statement.
02

Express the given identity in terms of tangent

The provided identity is \( \tan 3 u = \frac{\tan u\left(3-\tan ^{2} u\right)}{1-3 \tan ^{2} u} \). We need to verify that this expression equals the triple angle identity we recalled in the previous step.
03

Simplify the expression on the right

The expression \( \frac{\tan u (3 - \tan^2 u)}{1 - 3 \tan ^{2} u} \) can be expanded to \( \frac{3\tan u - \tan^3 u}{1 - 3 \tan ^{2} u} \). Notice this is the same as the triple angle formula for tangent \( \frac{3\tan u - \tan^3 u}{1 - 3 \tan^2 u} \).
04

Confirm both expressions are identical

Since the simplified right-hand side of the provided identity matches the standard triple angle identity formula for tangent, the expressions are indeed identical.

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Key Concepts

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

Tangent Functions
Tangent is a fundamental trigonometric function that is often used in various mathematical calculations. It relates the angles and ratios of triangles. In essence, the tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side.
  • Mathematically, it's expressed as: \( \tan(u) = \frac{\text{opposite}}{\text{adjacent}} \).
  • Tangent functions also have periodic properties, meaning they repeat values in a predictable pattern over intervals.
Tangent functions are essential in solving real-world problems involving angles and distances. They are also crucial in trigonometric identities and equations.
Understanding tangent functions includes knowing about their periodicity and asymptotes, which are values where the function tends to infinity.
Triple Angle Identities
Triple angle identities are trigonometric equations that involve angles that are three times larger than a given angle. They allow us to express functions of triple angles in terms of single angles. For tangent, the triple angle identity is:
  • \( \tan(3u) = \frac{3 \tan(u) - \tan^3(u)}{1 - 3 \tan^2(u)} \).
  • These identities show how complex trigonometric expressions can be simplified.
Triple angle identities are useful in simplifying equations where angles increase by multiples. They help in reducing the complexity of calculations, especially in solving trig equations and verifying identities. Understanding how to use these identities allows for more straightforward problem-solving.
Identity Verification
Identity verification in trigonometry involves showing that two different expressions are equivalent. It is an essential skill in understanding and applying trigonometric identities effectively. In the exercise, we verified the tangent identity by showing equivalence:
  • First, recall the standard form of the identity you're verifying.
  • Next, re-examine the given expression and try to transform it to match the standard form.
In the specific problem, both the given expression and the known triple angle identity for tangent were manipulated to show they are the same.
Verifying identities ensures an understanding of the identities themselves and strengthens problem-solving skills, offering confidence in their application. Being methodical and precise in these steps leads to successful identity verification.

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Most popular questions from this chapter

Use the graph of \(f\) to find the simplest expression \(g(x)\) such that the equation \(f(x)=g(x)\) is an identity. Verify this identity. $$f(x)=\frac{\sin x-\sin ^{3} x}{\cos ^{4} x+\cos ^{2} x \sin ^{2} x}$$

Because planets do not move in precisely circular orbits, the computation of the position of a planet equires the solution of Kepler's equation. Kepler's equation cannot be solved algebraically. It has the form \(M=\theta+e \sin \theta,\) where \(M\) is the mean anomaly, \(e\) is the eccentricity of the orbit, and \(\theta\) is an angle called the ecceneric anomaly. For the specified values of \(M\) and \(e\), use graphacal techniques to solve Kepler's equation for \(\theta\) to three edecimal places. Position of Pluto \(\quad M=0.09424, \quad e=0.255\)

Because planets do not move in precisely circular orbits, the computation of the position of a planet equires the solution of Kepler's equation. Kepler's equation cannot be solved algebraically. It has the form \(M=\theta+e \sin \theta,\) where \(M\) is the mean anomaly, \(e\) is the eccentricity of the orbit, and \(\theta\) is an angle called the ecceneric anomaly. For the specified values of \(M\) and \(e\), use graphacal techniques to solve Kepler's equation for \(\theta\) to three edecimal places. Position of Mars \(\quad M=4.028, \quad e=0.093\)

(a) Compare the decimal approximations of both sides of equation (1). (b) Find the acute angle \(\alpha\) such that equation ( 2) is an identity. (c) How does equation ( 1) relate to equation ( 2 )? (1) \(\sin 63^{\circ}-\sin 57^{\circ}=\sin 3^{\circ}\) (2) \(\sin (\alpha+\beta)-\sin (\alpha-\beta)=\sin \beta\)

Graphically solve the trigonometric equation on the indicated interval to two decimal places. $$\tan \left(\frac{1}{2} x+1\right)=\sin \frac{1}{2} x, \quad[-2 \pi, 2 \pi]$$
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