Textbook Question
In Exercises 35–38, use the power-reducing formulas to rewrite each expression as an equivalent expression that does not contain powers of trigonometric functions greater than 1.sin² x cos² x
530
views
6. Trigonometric Identities and More Equations
Introduction to Trigonometric Identities
3:41 minutesProblem 47
Textbook Question
In Exercises 47–54, use the figures to find the exact value of each trigonometric function.θsin -------2
Verified step by step guidance1
Identify the given angle \( \theta \) and the trigonometric function you need to find, which is \( \sin \frac{\theta}{2} \).
Recall the half-angle identity for sine: \( \sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}} \).
Determine the value of \( \cos \theta \) using the given figure or information.
Substitute the value of \( \cos \theta \) into the half-angle identity formula.
Decide the sign (positive or negative) of \( \sin \frac{\theta}{2} \) based on the quadrant in which \( \frac{\theta}{2} \) lies.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
3mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Functions
Trigonometric functions, such as sine, cosine, and tangent, relate the angles of a triangle to the lengths of its sides. The sine function, for example, is defined as the ratio of the length of the opposite side to the hypotenuse in a right triangle. Understanding these functions is essential for solving problems involving angles and distances.
Recommended video:
6:04
Introduction to Trigonometric Functions
Unit Circle
The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It provides a geometric representation of the trigonometric functions, where the x-coordinate corresponds to the cosine of an angle and the y-coordinate corresponds to the sine. Familiarity with the unit circle helps in determining the exact values of trigonometric functions for common angles.
Recommended video:
06:11Introduction to the Unit Circle
Exact Values of Trigonometric Functions
Exact values of trigonometric functions refer to the precise values of sine, cosine, and tangent for specific angles, often expressed in terms of square roots. For example, sin(30°) equals 1/2, and cos(45°) equals √2/2. Knowing these exact values is crucial for solving trigonometric equations and simplifying expressions without relying on calculators.
Recommended video:
6:04Introduction to Trigonometric Functions
6:19mWatch next
Master Even and Odd Identities with a bite sized video explanation from Patrick
Start learningRelated Videos
Related Practice