Find the exact value of each expression.
sin (13π/12)

Guida verificata passo dopo passo

Recognize that the angle \( \frac{13\pi}{12} \) is not one of the standard angles on the unit circle, so we need to express it as a sum or difference of angles whose sine values we know exactly.

Rewrite \( \frac{13\pi}{12} \) as \( \pi + \frac{\pi}{12} \) or as a sum of two angles such as \( \frac{3\pi}{4} + \frac{\pi}{3} \) or \( \pi - \frac{\pi}{12} \). For this problem, use the sum \( \frac{3\pi}{4} + \frac{\pi}{3} \) because both \( \frac{3\pi}{4} \) and \( \frac{\pi}{3} \) are standard angles.

Apply the sine addition formula: \( \sin(a + b) = \sin a \cos b + \cos a \sin b \). Here, \( a = \frac{3\pi}{4} \) and \( b = \frac{\pi}{3} \).

Substitute the known exact values: \( \sin \frac{3\pi}{4} = \frac{\sqrt{2}}{2} \), \( \cos \frac{3\pi}{4} = -\frac{\sqrt{2}}{2} \), \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \), and \( \cos \frac{\pi}{3} = \frac{1}{2} \).

Combine these values into the formula: \( \sin \left( \frac{3\pi}{4} + \frac{\pi}{3} \right) = \sin \frac{3\pi}{4} \cos \frac{\pi}{3} + \cos \frac{3\pi}{4} \sin \frac{\pi}{3} \), then simplify the expression step-by-step to find the exact value.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Unit Circle and Radian Measure

The unit circle represents angles in radians, where 2π radians equal 360 degrees. Understanding how to locate angles like 13π/12 on the unit circle helps in determining the sine value by relating it to known reference angles.

Video consigliato:

Percorso guidato

06:11

Introduction to the Unit Circle

Angle Sum and Difference Identities

These identities allow the calculation of trigonometric functions for angles expressed as sums or differences of standard angles. For example, sin(13π/12) can be rewritten as sin(π + π/12) or sin(3π/4 + π/6) to use known sine and cosine values.

Video consigliato:

Percorso guidato

2:25

Verifying Identities with Sum and Difference Formulas

Exact Values of Sine and Cosine for Special Angles

Certain angles like π/6, π/4, and π/3 have well-known exact sine and cosine values. Using these values in combination with angle sum or difference formulas enables finding the exact sine of non-standard angles such as 13π/12.

Video consigliato:

Percorso guidato