which cannot be the value of sine for
my
values of Q
Answers
Step-by-step explanation:
triangles. Then we’ll use these exact values to answer the above challenges.
sin 45°: You may recall that an isosceles right triangle with sides of 1 and with hypotenuse of square root of 2 will give you the sine of 45 degrees as half the square root of 2.
sin 30° and sin 60°: An equilateral triangle has all angles measuring 60 degrees and all three sides are equal. For convenience, we choose each side to be length 2. When you bisect an angle, you get 30 degrees and the side opposite is 1/2 of 2, which gives you 1. Using that right triangle, you get exact answers for sine of 30°, and sin 60° which are 1/2 and the square root of 3 over 2 respectively.
Using these results – sine 15°
How do you find the value of the sine of 15°?
Sine of half an angle in the first quadrant is given by the expression:
So the sine of 1/2 of 30° will be:
which gives us
or
Note: We could also find the sine of 15 degrees using sine (45° − 30°).
sin 75°: Now using the formula for the sine of the sum of 2 angles,
sin(A + B) = sin A cos B + cos A sin B,
we can find the sine of (45° + 30°) to give sine of 75 degrees.
We now find the sine of 36°, by first finding the cos of 36°.
cos 36°: The cosine of 36 degrees can be calculated by using a pentagon. See cos36° at CutTheKnot where it is shown that
Putting these values on a right triangle and solving for the unknown side, we can conclude:
sin 18°: Now, the sine of 18 degrees comes from the sine of half of 36 degrees.
Calculating this, the sine of 18 degrees becomes
sin 3°: The above leads you to one of the paths to sine of 3 degrees and to sine of 6 degrees.
For example, sine (18° - 15°) will give us the sine of 3 degrees. which is
sin 3° = sin (18° − 15°) = sin 18° cos 15° − sin 15° cos 18°
This gives us the following value of sin 3°:
or other forms depending how you factor the above.
sin 6°: Using the above, one can compute the sine of 6 degrees finally as sine of twice 3 degrees to arrive at
sin 18° and sin 72°: Taking the equivalent sine and cosine values of 15° and 18° on the right hand side of
sin 3° = sin (18° − 15°) = sin 18° cos 15° − sin 15° cos 18°
gives us:
sin 3° = sin 18° sin 75° − sin 15° sin 72°
We can calculate the values of the sines of 18° and 72° from the above expression.
Sines of other angles
Many angles can be computed exactly by many methods. Another practical formula is the sine of 3 times an angle:
sin 3A = 3 sin A − 4 sin3A
sin 9°: For example, the sine of 9 degrees is the sine of (3×3°).
So, with A = 3, we arrive at
And so on.
sin 1°: Now, to find the sine of one degree, one needs to know sine of one third of three degrees!
One needs to solve the above for sin (A) in terms of 3A, and this involves solving the cubic. As you know, the cubic was solved many, many years ago.
There are three solutions and one needs to know which one to use and when! Experience has taught me to use the following for a quadrant I angle (the "I" in this expression stands for the imaginary number √(−1). See Complex Numbers for more information.)
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Use the following when you have a quadrant II angle:
Use the following for quadrant III angles:
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So, the expression for sine(1°) becomes
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