Evaluating the Trigonometric Expression sin 300° and cos 150°
Understanding and evaluating trigonometric expressions is a fundamental skill in mathematics. In this article, we will explore how to evaluate sin 300° and cos 150° using trigonometric identities and formulas. We will take a step-by-step approach to solve the expression sin 300° - cos 150°.
Trigonometric Identities and Formulas
The problem of evaluating sin 300° - cos 150° can be approached using several trigonometric identities and formulas. Some of the key identities that we will use are:
sin(α β) sin α cos β cos α sin β cos(α - β) cos α cos β sin α sin β sin(α - 90°) -cos α cos(α 90°) -sin αEvaluating sin 300°
To evaluate sin 300°, we need to understand the position of 300° in the unit circle. This angle is in the fourth quadrant where the sine function is negative. Using the trigonometric identity:
sin 300° sin(360° - 60°) -sin 60°
Since sin 60° √3/2, we have:
sin 300° -√3/2
Evaluating cos 150°
Similarly, to evaluate cos 150°, we need to look at the angle's position in the unit circle. This angle is in the second quadrant where the cosine function is negative. Using the trigonometric identity:
cos 150° cos(180° - 30°) -cos 30°
Since cos 30° √3/2, we have:
cos 150° -√3/2
Evaluating the Expression sin 300° - cos 150°
Now, we will evaluate the expression sin 300° - cos 150°:
sin 300° - cos 150° -√3/2 - (-√3/2) -√3/2 √3/2 0
This simplifies to 0, as seen above. The positive and negative terms cancel each other out, resulting in 0.
To further cement this understanding, let's use the trigonometric identities to express sin 300° and cos 150° differently:
sin 300° sin(300°) sin(360° - 60°) sin(60°) cos(300°) cos(60°) sin(300°)
Since sin(300°) -sin(60°) and cos(60°) 1/2 and sin(300°) -√3/2, we get:
sin(300°) -√3/2
cos 150° cos(150°) cos(180° - 30°) -cos(30°)
Since cos(30°) √3/2, we get:
cos(150°) -√3/2
Subtracting these values, we get:
sin(300°) - cos(150°) -√3/2 - (-√3/2) 0
This confirms our earlier result that the expression evaluates to 0.
Conclusion
In summary, we have evaluated the expression sin 300° - cos 150° using various trigonometric identities and formulas. We demonstrated that this expression simplifies to 0, confirming our initial result.