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Chapter 3: Problem 47

Write each expression as a function of \(\alpha\) alone. \(\tan \left(180^{\circ}+\alpha\right)\)

Short Answer

Expert verified
\(\tan(180^{\text{掳}} + \thinspace\text{伪}) = \tan(\thinspace\text{伪})\)

Step by step solution

01

- Understand the Trigonometric Identity

Recall the trigonometric identity for tangent function with a sum: \(\tan(180^{\text{掳}} + \theta) = \tan(\theta)\). This is because: \[\tan(180^{\text{掳}} + \theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{-\text{opposite}}{-\text{adjacent}} = \frac{\text{opposite}}{\text{adjacent}} = \tan(\theta) \]
02

- Apply the Identity

Apply the identity to the given expression: \(\tan(180^{\circ}+\theta) = \tan(\theta)\)Replace \(\theta\) with \(\alpha\) to get: \(\tan(180^{\circ}+\thinspace\alpha) = \tan(\alpha)\)

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Key Concepts

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

tangent function
Tangent is one of the primary functions in trigonometry, usually abbreviated as \tan\. The tangent of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. This relationship can be written as: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]. You will often see the tangent function used in various trigonometric identities and problems.
angle sum identities
Angle sum identities help simplify expressions in trigonometry. One essential identity involves the tangent function for an angle sum: \( \tan(\alpha + \beta) \). This is formulated as: \[ \tan(\alpha + \beta) = \frac{\tan(\alpha) + \tan(\beta)}{1 - \tan(\alpha)\tan(\beta)} \]. In our exercise, we specifically use the identity for \( \tan(180^{\circ} + \alpha) \). Due to the periodic properties of the tangent function, this simplifies to \( \tan(\alpha) \). The reason behind this is grounded in the trigonometric circle and symmetry: \[ \tan(180^{\circ} + \theta) = \tan(\theta) \].
trigonometric simplification
Trigonometric simplification often involves using identities to reduce complex expressions to simpler forms. For example, the angle sum identity for tangent lets us express \( \tan(180^{\circ} + \alpha) \) in terms of \( \alpha \) alone. By recognizing and applying these identities:
  • We can transform the given expression into a simpler one.
  • We aim for an expression that uses fewer terms and standard angles.
This process is crucial in trigonometry for easier computation and understanding relationships between different angles.

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Most popular questions from this chapter

Match each expression with an equivalent expression from \((a)-(h) .\) Do not use a calculator: a. \(\csc (\pi / 3)\) b. \(\sec (3 \pi / 8)\) c. \(\sin \left(0^{\circ}\right)\) d. \(\tan \left(50^{\circ}\right)\) e. \(\cos \left(70^{\circ}\right)\) f. \(\tan (\pi / 6)\) g. \(\cos (\pi / 12)\) h. \(\cot \left(5^{\circ}\right)\) $$ \sec (\pi / 6) $$

Write each expression as a function of \(\alpha\) alone. $$ \cos (\alpha-\pi / 2) $$

Complete the sum and difference identities for sine. a. \(\sin (x+y)= \)______ b. \(\sin (x-y)=\) ______

Prove that each equation is an identity: $$ \cos ^{2}(A-B)-\cos ^{2}(A+B)=\sin ^{2}(A+B)-\sin ^{2}(A-B) $$

In each case, find \(\sin \alpha, \cos \alpha, \tan \alpha, \csc \alpha, \sec \alpha,\) and \(\cot \alpha\). \(\cos 2 \alpha=1 / 3\) and \(360^{\circ}<2 \alpha<450^{\circ}\)
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