Understanding 2 sin 15° cos 15° Without a Calculator
In this article, we will explore how to simplify u200B2 sin 15 u00B0 cos 15 u00B0 by using trigonometric identities. Specifically, we will use the double angle identity for sine to find the exact value.
Step-by-Step Guide to Simplification
The double angle identity for sine is a powerful tool in trigonometry. It states that:
sin 2u03B8 2 sin u03B8 cos u03B8
This identity can be used to simplify expressions involving the product of sine and cosine functions of the same angle. Here, we are given the expression u200B2 sin 15 u00B0 cos 15 u00B0. Let's apply the double angle identity step-by-step.
Applying the Double Angle Identity
Let u03B8 15 u00B0. Then:
Substitute u03B8 15 u00B0 into the double angle identity: u200B2 sin 15 u00B0 cos 15 u00B0 sin (2 u03B8) Simplify the expression: u200Bsin (2 u03B8) sin (2 u00D7 15 u00B0) sin 30 u00B0Evaluating sin 30°
Now, we need to evaluate u200Bsin 30 u00B0. There are several ways to do this:
Using the 30-60-90 Triangle: In a 30-60-90 triangle, the sine of 30° is the ratio of the length of the side opposite the 30° angle to the hypotenuse. If we consider a 30-60-90 triangle with the hypotenuse as 2 and the side opposite the 30° angle as 1, then: u200Bsin 30 u00B0 u200B1/2 Using the Unit Circle: On the unit circle, the sine of 30° corresponds to the y-coordinate of the point where the terminal side of the angle 30° intersects the circle. This point is (u221A3/2, 1/2), giving us the sine value of 1/2.Final Answer
Therefore, we can conclude:
u200B2 sin 15 u00B0 cos 15 u00B0 sin 30 u00B0 u200B1/2
Conclusion
Understanding how to use trigonometric identities like the double angle identity for sine can be very useful in simplifying complex expressions and solving trigonometric problems without a calculator. This method not only helps in simplification but also in developing a deeper understanding of trigonometric functions.