How do you find the exact value of sin(-105) degrees without a calculator
-105º is a 75º reference angle in Quadrant III
75º = 45º + 30º
using the trig identity ... sin(α + β) = sin(α) cos(β) + cos(α) sin(β)
I'm sorry I'm stupid could you please simplify that, where would the 45 and 30 go? Thank you for your help
What R_scott is implying is
sin(-105°)
= sin(255°) , -105° and 255 are coterminal angles
= - sin 75°
= - sin(45 + 30)
= - (sin45 cos30 + cos45 sin30)
we both would assume that you know the exact value of these standard angles
sin75º = sin(45º + 30º)
= sin45º cos30º + cos45º sin30º
= √2/2 * √3/2 + √2/2 * 1/2
= √6/4 + √2/4
= √2/4 (√3 + 1)
Now use your reference angle of 75º in QIII and you have
sin -105º = -sin75º
To find the exact value of sin(-105) degrees without a calculator, we can use the properties of the sine function and trigonometric identities. Here's how:
1. Start by converting -105 degrees to the equivalent angle within the range of 0 to 360 degrees. Since the sine function has a period of 360 degrees, we can add or subtract multiples of 360 to the given angle to get an equivalent angle within this range. For -105 degrees, we can add 360 to it multiple times until we get an angle between 0 and 360 degrees.
-105 + 360 = 255 (within the range of 0 to 360 degrees)
2. Use the angle 255 degrees to find the exact value of sin(255) degrees.
3. Determine the reference angle: Since the sine function is positive in both Quadrant 1 and Quadrant 2, we need to find the reference angle in Quadrant 2 that has the same sine value as our angle of 255 degrees.
The reference angle is found by subtracting the angle from 180 degrees:
Reference angle = 180 - 255 = -75 degrees
4. Determine the sign: Since the sine function is positive in both Quadrant 1 and Quadrant 2, the sign of the sine value will be positive.
5. Use a known exact value: sin(-75) degrees is equivalent to sin(-75 + 180) degrees since adding or subtracting multiples of 180 degrees does not change the value of the sine function. Therefore, sin(-75) degrees is equivalent to sin(105) degrees.
6. Use a trigonometric identity: To find the exact value of sin(105) degrees, we can use the identity sin(A) = sin(180 - A). Therefore, sin(105) degrees = sin(180 - 105) degrees.
sin(180 - 105) degrees = sin(75) degrees
7. Use the known value from the unit circle: By referring to the unit circle or the values of common angles, we know that sin(75) degrees is equal to (√6 - √2) / 4.
Therefore, the exact value of sin(-105) degrees without a calculator is (√6 - √2) / 4.