Textbook Question
Find the exact value of each expression.
sin (13π/12)
596
views
6. Trigonometric Identities and More Equations
Sum and Difference Identities
Problem 5.18
Textbook Question
Find the exact value of each expression.
sin (π/12)
Verified step by step guidance1
Recognize that \( \frac{\pi}{12} \) is an angle that can be expressed as a difference of two special angles: \( \frac{\pi}{4} - \frac{\pi}{6} \).
Use the sine difference identity: \( \sin(a - b) = \sin a \cos b - \cos a \sin b \).
Substitute \( a = \frac{\pi}{4} \) and \( b = \frac{\pi}{6} \) into the identity: \( \sin(\frac{\pi}{4} - \frac{\pi}{6}) = \sin(\frac{\pi}{4}) \cos(\frac{\pi}{6}) - \cos(\frac{\pi}{4}) \sin(\frac{\pi}{6}) \).
Recall the exact values: \( \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \), \( \cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2} \), \( \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \), and \( \sin(\frac{\pi}{6}) = \frac{1}{2} \).
Substitute these values into the expression: \( \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2} \).
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.
Unit Circle
The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is fundamental in trigonometry as it provides a geometric interpretation of the sine, cosine, and tangent functions. The coordinates of points on the unit circle correspond to the cosine and sine values of angles measured in radians, making it essential for evaluating trigonometric functions like sin(π/12).
Recommended video:
06:11
Introduction to the Unit Circle
Angle Sum and Difference Identities
Angle sum and difference identities are formulas that express the sine, cosine, and tangent of the sum or difference of two angles in terms of the sine and cosine of those angles. For example, sin(a ± b) = sin(a)cos(b) ± cos(a)sin(b). These identities are particularly useful for finding the exact values of trigonometric functions for angles that are not standard, such as π/12, by expressing them as the sum or difference of known angles.
Recommended video:
2:25Verifying Identities with Sum and Difference Formulas
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 instance, sin(π/12) can be calculated using known angles like π/6 and π/4. Understanding how to derive these exact values using identities and the unit circle is crucial for solving trigonometric problems accurately.
Recommended video:
6:04Introduction to Trigonometric Functions
6:14mWatch next
Master Sum and Difference of Sine & Cosine with a bite sized video explanation from Patrick
Start learningRelated Videos
Related Practice