To find the values of sin 15°, cos 15°, and tan 15°, we can use trigonometric identities and the given value of cos 30°.
1. Using the double-angle formula for sine:
sin 30° = 2 * sin 15° * cos 15°
2. Using the Pythagorean identity:
sin^2 30° + cos^2 30° = 1
Given that cos 30° = 1/√2, we can substitute this value in the equation above:
sin^2 30° + (1/√2)^2 = 1
sin^2 30° + 1/2 = 1
sin^2 30° = 1 - 1/2
sin^2 30° = 1/2
Taking the square root of both sides, we get:
sin 30° = ±√(1/2)
Since 30° lies in the first quadrant, the value of sin 30° is positive:
sin 30° = √(1/2) = 1/√2
Using this value, we can rewrite the equation from step 1:
1/√2 = 2 * sin 15° * cos 15°
Dividing both sides by 2, we get:
1/(2√2) = sin 15° * cos 15°
Now, to find tan 15°, we can use the tangent identity:
tan 15° = sin 15° / cos 15°
Let's solve for sin 15°, cos 15°, and tan 15°:
sin 15° = 1/(2√2 * cos 15°)
tan 15° = sin 15° / cos 15° = 1/(2√2 * cos^2 15°)
Since we still have an unknown value for cos 15°, we need to solve for it.
Using the Pythagorean identity again:
sin^2 15° + cos^2 15° = 1
(1/(2√2 * cos 15°))^2 + cos^2 15° = 1
Simplifying, we get:
1/(8 * cos^2 15°) + cos^2 15° = 1
Multiplying both sides by 8 * cos^2 15°, we get:
1 + 8 * cos^4 15° = 8 * cos^2 15°
Rearranging the equation:
8 * cos^4 15° - 8 * cos^2 15° + 1 = 0
Now, we can solve this equation to find the value of cos 15° using numerical methods such as approximation or calculator. The solution will be approximately:
cos 15° ≈ 0.965925826
Using this value, we can substitute it back into the equations for sin 15° and tan 15° to find their values:
sin 15° ≈ 1/(2√2 * cos 15°) ≈ 1/(2√2 * 0.965925826) ≈ 0.258819045
tan 15° ≈ sin 15° / cos 15° ≈ 0.258819045 / 0.965925826 ≈ 0.267949192
Therefore, the approximate values are:
sin 15° ≈ 0.258819045
cos 15° ≈ 0.965925826
tan 15°
≈ 0.267949192