Textbook Question
Write each function value in terms of the cofunction of a complementary angle.
tan 174° 03'
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6. Trigonometric Identities and More Equations
Sum and Difference Identities
Problem 37
Textbook Question
Find one value of θ or x that satisfies each of the following.
tan θ = cot(45° + 2θ)
Verified step by step guidance1
Recall the definition of cotangent in terms of tangent: \(\cot \alpha = \frac{1}{\tan \alpha}\). So the equation \(\tan \theta = \cot(45^\circ + 2\theta)\) can be rewritten as \(\tan \theta = \frac{1}{\tan(45^\circ + 2\theta)}\).
Multiply both sides of the equation by \(\tan(45^\circ + 2\theta)\) to get rid of the fraction: \(\tan \theta \cdot \tan(45^\circ + 2\theta) = 1\).
Use the tangent addition formula to express \(\tan(45^\circ + 2\theta)\): \(\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}\). Here, \(a = 45^\circ\) and \(b = 2\theta\), so \(\tan(45^\circ + 2\theta) = \frac{\tan 45^\circ + \tan 2\theta}{1 - \tan 45^\circ \tan 2\theta}\).
Substitute \(\tan 45^\circ = 1\) into the expression to simplify: \(\tan(45^\circ + 2\theta) = \frac{1 + \tan 2\theta}{1 - \tan 2\theta}\).
Replace \(\tan(45^\circ + 2\theta)\) in the equation \(\tan \theta \cdot \tan(45^\circ + 2\theta) = 1\) with the simplified expression and solve for \(\theta\): \(\tan \theta \cdot \frac{1 + \tan 2\theta}{1 - \tan 2\theta} = 1\). From here, you can proceed to isolate \(\theta\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Relationship Between Tangent and Cotangent
Tangent and cotangent are reciprocal trigonometric functions, where cot(α) = 1/tan(α). Understanding this relationship allows us to rewrite cotangent expressions in terms of tangent, facilitating equation solving.
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Angle Sum Identities
Angle sum identities express trigonometric functions of sums of angles, such as tan(A + B) = (tan A + tan B) / (1 - tan A tan B). These identities help simplify or transform expressions involving sums like 45° + 2θ.
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Solving Trigonometric Equations
Solving trigonometric equations involves manipulating expressions using identities and algebraic techniques to isolate the variable. Recognizing equivalent angles and periodicity is key to finding valid solutions.
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