find exact values sin 60 cos 30 tan 45 cos 45

find exact values sin 60 cos 30 tan 45 cos 45

The Correct Answer and Explanation is:

Let’s find the exact values for the following trigonometric functions:

1. sin 60°
2. cos 30°
3. tan 45°
4. cos 45°

1. sin 60°

To find sin 60°, we can use the known values from trigonometric ratios for special angles. Specifically, 60° is one of the standard angles in trigonometry.
We know that:sin⁡60∘=32\sin 60^\circ = \frac{\sqrt{3}}{2}sin60∘=23​​

This is derived from an equilateral triangle where each angle is 60°. By dividing the equilateral triangle in half, you get a 30°-60°-90° triangle. The sides of the 30°-60°-90° triangle are in the ratio 1:√3:2, with the hypotenuse being 2. The opposite side to the 60° angle is √3, so:sin⁡60∘=oppositehypotenuse=32\sin 60^\circ = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\sqrt{3}}{2}sin60∘=hypotenuseopposite​=23​​

2. cos 30°

Next, for cos 30°, we use the property of complementary angles, noting that:cos⁡30∘=sin⁡60∘=32\cos 30^\circ = \sin 60^\circ = \frac{\sqrt{3}}{2}cos30∘=sin60∘=23​​

This is because of the identity cos⁡(90∘−θ)=sin⁡θ\cos(90^\circ – \theta) = \sin \thetacos(90∘−θ)=sinθ. Hence, cos 30° is also equal to sin 60°.

3. tan 45°

The tangent of 45° is another standard value. In a right-angled isosceles triangle (45°-45°-90°), the two legs are of equal length. The tangent is the ratio of the opposite side to the adjacent side. Since both sides are equal in this triangle, the tangent of 45° is:tan⁡45∘=1\tan 45^\circ = 1tan45∘=1

4. cos 45°

Finally, for cos 45°, we again refer to the 45°-45°-90° triangle. The sides are in the ratio 1:1:√2. The cosine is the ratio of the adjacent side to the hypotenuse:cos⁡45∘=12=22\cos 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}cos45∘=2​1​=22​​

Conclusion

• sin 60° = 32\frac{\sqrt{3}}{2}23​​
• cos 30° = 32\frac{\sqrt{3}}{2}23​​
• tan 45° = 1
• cos 45° = 22\frac{\sqrt{2}}{2}22​​

These values come from known trigonometric ratios for special angles and right triangles. The key is recognizing the relationships between the angles and the sides of these special triangles.