Find the exact value of

1. tan 75 degrees
2. Cos 15 degrees
3. Sin 105 degrees

you know the values for 30 and 45 degrees

use your sum/difference and half-angle formulas

75=45+30
15=30/2
105=90+15

Tan 75 = tan (45 +30)

= tan 45 + tan 30/1-tan 45 *tan 30
= (1+ 1/√3)/(1-1/√3) = √3 + 1/√3-1
= √3 + 1/√3-1 (afterationalization )
= 2+√3
cos 15 = cos (45 - 30)

= cos 45 cos 30+ sin 45 sin 30
=

1. To find the exact value of tan 75 degrees, we can use the tangent addition formula:

tan (α + β) = (tan α + tan β) / (1 - tan α * tan β)

By substituting α = 45 degrees and β = 30 degrees, and using the values of tangent for 45 and 30 degrees, we can find the value of tan 75 degrees.

tan 75 degrees = (tan 45 degrees + tan 30 degrees) / (1 - tan 45 degrees * tan 30 degrees)

tan 45 degrees = 1 and tan 30 degrees is 1/√3.
Substitute these values into the equation:

tan 75 degrees = (1 + 1/√3) / (1 - 1 * 1/√3)

Rationalize the denominator by multiplying the expression by (√3 + 1) / (√3 + 1):

tan 75 degrees = (1 + 1/√3) * (√3 + 1) / (1 - 1 * 1/√3) * (√3 + 1)

Simplify the expression:

tan 75 degrees = (√3 + 1 + 1/√3 + 1/√3√3) / (1 - 1/√3)

tan 75 degrees = (√3 + 1√3 + 1 + 1√3) / (√3 - 1√3)

tan 75 degrees = 2√3 + √3 / 3 - √3

tan 75 degrees = √3

Therefore, the exact value of tan 75 degrees is √3.

2. To find the exact value of cos 15 degrees, we can use the half-angle formula for cosine:
cos (α/2) = ±√(1 + cos α) / 2

By substituting α = 30 degrees, we can find the value of cos 15 degrees.
cos 30 degrees = √3/2
Substitute this value into the half-angle formula:

cos 15 degrees = ±√(1 + √3/2) / 2

Simplify the expression:

cos 15 degrees = ±√(2 + √3) / 2

Therefore, the exact value of cos 15 degrees is ±√(2 + √3) / 2.

3. To find the exact value of sin 105 degrees, we can use the difference of angles formula for sine:
sin (α - β) = sin α * cos β - cos α * sin β

By substituting α = 120 degrees and β = 15 degrees, and using the values of sine and cosine for 120 and 15 degrees, we can find the value of sin 105 degrees.

sin 120 degrees = √3/2
cos 15 degrees = ±√(2 + √3) / 2

Substitute these values into the formula:

sin 105 degrees = sin (120 degrees - 15 degrees) = sin 120 degrees * cos 15 degrees - cos 120 degrees * sin 15 degrees

sin 105 degrees = (√3/2) * (±√(2 + √3) / 2) - (-1/2) * (±√(2 + √3) / 2)

Simplify the expression:

sin 105 degrees = ±√3 * √(2 + √3) / 4 + √(2 + √3) / 4

sin 105 degrees = (√3 + 1) * √(2 + √3) / 4

Therefore, the exact value of sin 105 degrees is (√3 + 1) * √(2 + √3) / 4.

To find the exact value of the trigonometric functions, we can use special angles and trigonometric identities. Let's calculate each one step-by-step:

1. To find the value of tan 75 degrees:

We know that tan(x) = sin(x) / cos(x).

Using the double-angle formula for sin(2x), we can rewrite sin 150 degrees as:
sin 150 degrees = sin(2 * 75 degrees) = 2 * sin 75 degrees * cos 75 degrees.

We also know that cos 2x = cos^2(x) - sin^2(x), so we can rewrite cos(2 * 75 degrees) as:
cos(150 degrees) = cos^2(75 degrees) - sin^2(75 degrees).

Now, let's substitute in the values we already have:
2 * sin 75 degrees * cos 75 degrees = cos^2(75 degrees) - sin^2(75 degrees).

Let's solve for cos^2(75 degrees) - sin^2(75 degrees):
2 * sin 75 degrees * cos 75 degrees = cos^2(75 degrees) - sin^2(75 degrees),
2 * sin 75 degrees * cos 75 degrees = cos^2(75 degrees) - (1 - cos^2(75 degrees)),
2 * sin 75 degrees * cos 75 degrees = cos^2(75 degrees) - 1 + cos^2(75 degrees),
2 * sin 75 degrees * cos 75 degrees = 2 * cos^2(75 degrees) - 1.

Now it becomes a quadratic equation:
2 * sin 75 degrees * cos 75 degrees - 2 * cos^2(75 degrees) = -1,
2 * sin 75 degrees * cos 75 degrees - 2 * cos^2(75 degrees) + 1 = 0.

Using the quadratic formula:
cos^2(75 degrees) = (-b ± √(b^2 - 4ac)) / 2a,

where a = 2, b = -2 * sin 75 degrees, and c = 1, we plug in the values and solve for cos^2(75 degrees):
cos^2(75 degrees) = (-(-2 * sin 75 degrees) ± √((2 * sin 75 degrees)^2 - 4 * 2 * 1)) / (2 * 2),
cos^2(75 degrees) = (2 * sin 75 degrees ± √(4 sin^2(75 degrees) - 8)) / 4.

Since cos^2(75 degrees) is positive, we only consider the positive square root:
cos^2(75 degrees) = (2 * sin 75 degrees + √(4 sin^2(75 degrees) - 8)) / 4.

Therefore, cos(75 degrees) is equal to (2 * sin 75 degrees + √(4 sin^2(75 degrees) - 8)) / 2.

Now, using the identity tan(x) = sin(x) / cos(x), we can find tan(75 degrees):
tan 75 degrees = sin 75 degrees / cos 75 degrees,
tan 75 degrees = sin 75 degrees / ((2 * sin 75 degrees + √(4 sin^2(75 degrees) - 8)) / 2).

Simplifying further, we have:
tan 75 degrees = 2 * sin 75 degrees / (2 * sin 75 degrees + √(4 sin^2(75 degrees) - 8)).

2. To find the value of cos 15 degrees:

We can use the half-angle formula for cosine:
cos(2x) = 2 * cos^2(x) - 1.

Now, let's substitute 2x = 30 degrees (2 * 15 degrees):
cos(30 degrees) = 2 * cos^2(15 degrees) - 1.

Let's solve for cos^2(15 degrees):
cos^2(15 degrees) = (cos(30 degrees) + 1) / 2.

Therefore, cos(15 degrees) is equal to the square root of ((cos(30 degrees) + 1) / 2).

3. To find the value of sin 105 degrees:

We know that sin(180 degrees - x) = sin(x), so we can rewrite:
sin 105 degrees = sin(180 degrees - 105 degrees) = sin 75 degrees.

Using the value we calculated in question 1, we know that sin 75 degrees = tan 75 degrees * cos 75 degrees / 2.

Hence, sin 105 degrees = sin 75 degrees = tan 75 degrees * cos 75 degrees / 2.

Following these steps, we can find the exact value of tan 75 degrees, cos 15 degrees, and sin 105 degrees.