Textbook Question
Simplify each expression.
sin 158.2°/(1 + cos 158.2°)
681
views
6. Trigonometric Identities and More Equations
Introduction to Trigonometric Identities
Problem 5.42
Textbook Question
Simplify each expression.
±√[(1 - cos 5A)/(1 + cos 5A)]
Verified step by step guidance1
Recognize that the expression \( \pm \sqrt{\frac{1 - \cos 5A}{1 + \cos 5A}} \) can be simplified using trigonometric identities.
Recall the identity for tangent half-angle: \( \tan \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}} \).
Identify that the expression \( \pm \sqrt{\frac{1 - \cos 5A}{1 + \cos 5A}} \) matches the form of the tangent half-angle identity.
Conclude that \( \pm \sqrt{\frac{1 - \cos 5A}{1 + \cos 5A}} = \tan \frac{5A}{2} \).
Thus, the expression simplifies to \( \tan \frac{5A}{2} \).
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Was this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables. Key identities include the Pythagorean identities, angle sum and difference identities, and double angle formulas. Understanding these identities is crucial for simplifying trigonometric expressions, as they allow for the transformation of functions into more manageable forms.
Recommended video:
5:32
Fundamental Trigonometric Identities
Cosine Function Properties
The cosine function, denoted as cos(θ), represents the adjacent side over the hypotenuse in a right triangle. It has specific properties, such as being an even function (cos(-θ) = cos(θ)) and periodic with a period of 2π. Recognizing how cosine behaves under various transformations is essential for simplifying expressions that involve cosines, especially in the context of identities.
Recommended video:
5:53Graph of Sine and Cosine Function
Rationalizing Expressions
Rationalizing expressions involves eliminating radicals from the denominator or simplifying complex fractions. In trigonometry, this often means rewriting expressions to make them easier to work with or to reveal underlying relationships. This technique is particularly useful when dealing with square roots and can help in simplifying expressions like ±√[(1 - cos 5A)/(1 + cos 5A)].
Recommended video:
2:58Rationalizing Denominators
6:19mWatch next
Master Even and Odd Identities with a bite sized video explanation from Patrick
Start learningRelated Videos
Related Practice