Find The Exact Value Of Tan 15° Using The Half-Angle Identity. Tan 15°=
The correct answer and explanation is :
To find the exact value of tan(15°) using the half-angle identity, we start by recognizing that 15° is half of 30°. The half-angle identity for tangent is:
[ \tan\left(\frac{\theta}{2}\right) = \frac{1 – \cos(\theta)}{\sin(\theta)} ]
Setting (\theta = 30^\circ), we have:
[ \tan(15^\circ) = \tan\left(\frac{30^\circ}{2}\right) = \frac{1 – \cos(30^\circ)}{\sin(30^\circ)} ]
We know that:
- (\cos(30^\circ) = \frac{\sqrt{3}}{2})
- (\sin(30^\circ) = \frac{1}{2})
Substituting these values in:
[ \tan(15^\circ) = \frac{1 – \frac{\sqrt{3}}{2}}{\frac{1}{2}} ]
Simplify the numerator:
[ 1 – \frac{\sqrt{3}}{2} = \frac{2}{2} – \frac{\sqrt{3}}{2} = \frac{2 – \sqrt{3}}{2} ]
Now, divide by (\frac{1}{2}):
[ \tan(15^\circ) = \frac{\frac{2 – \sqrt{3}}{2}}{\frac{1}{2}} = 2 – \sqrt{3} ]
Therefore, the exact value of (\tan(15^\circ)) is (2 – \sqrt{3}).
Explanation:
The half-angle identities are useful for determining the trigonometric values of angles that are not commonly found on the unit circle. In this case, we used the half-angle identity for tangent to find (\tan(15^\circ)). By expressing 15° as half of 30°, we applied the identity to known values of sine and cosine at 30°. This method provides an exact value without resorting to decimal approximations.
Understanding and applying half-angle identities is crucial in trigonometry, especially when dealing with angles that are not standard on the unit circle. These identities allow us to break down complex angles into simpler components, facilitating exact calculations and deeper insights into trigonometric relationships.