How can I evaluate the expression below without using calculator?
cos15*2sin15
Thank you!
you can use the sum and difference rules
cos15 = cos (45-30) = cos45*cos30 + sin45*sin30
sin15 = sin (45-30) = sin45*cos30 - cos45*sin30
45 and 30 are the common trig values to know
I got sqrt6/4 - sqrt2/4 for sin15 and sqrt6/4 + sqrt2/4 for cos15. Is it right?
And the answer would be 6radical3/4???
Thank you!
Also, can you give me the exact formulas for the sum and difference rules (both sin and cos) without substituting any angles there? Thanks.
OR
cos15*2sin15
= 2(sin15)(cos15)
= sin 30
= 1/2
How did you do this, Reiny? I don't understand. How does 2(sin15)(cos15) equal sin30? Please help!
Thanks.
the "half-angle" formula
sin 2A = 2(sinA)(cosA)
in this case A = 15º
by the way, you can check on a calculator
your (cos15)(2sin15) = .5 or 1/2
To evaluate the expression "cos15 * 2sin15" without using a calculator, you can use some trigonometric identities and properties.
First, let's analyze the expression separately:
1. cos15: We know that the cosine function of an angle is equal to the adjacent side divided by the hypotenuse in a right triangle. However, 15 degrees is not one of the special angles, so we cannot directly determine the values of the sides. In this case, we can use the cosine addition formula:
cos(A + B) = cosAcosB - sinAsinB
We can rewrite cos15 as cos(45 - 30):
cos15 = cos(45 - 30)
Applying the cosine addition formula:
cos(45 - 30) = cos45cos30 + sin45sin30
We know that cos45 and sin45 are the same and equal to √2/2, and cos30 = √3/2, and sin30 = 1/2:
cos15 = (√2/2)(√3/2) + (√2/2)(1/2)
cos15 = √6/4 + √2/4
cos15 = (√6 + √2)/4
2. 2sin15: Similarly, we can apply the same cosine addition formula, but this time with the sine function.
sin(A + B) = sinAcosB + cosAsinB
We can rewrite sin15 as sin(45 - 30):
sin15 = sin(45 - 30)
Applying the sine addition formula:
sin(45 - 30) = sin45cos30 - cos45sin30
Since sin45 and cos45 are equal to √2/2, and cos30 = √3/2, and sin30 = 1/2:
sin15 = (√2/2)(√3/2) - (√2/2)(1/2)
sin15 = √6/4 - √2/4
sin15 = (√6 - √2)/4
Now, we can substitute these values back into the original expression and simplify:
cos15 * 2sin15 = [ (√6 + √2)/4 ] * [ 2 * (√6 - √2)/4 ]
Multiplying the numerators and denominators, we get:
[ (√6 + √2)(2√6 - 2√2) ] / (4 * 4)
Simplifying further:
[ 2(6) - 2(2) ] / 16
[ 12 - 4 ] / 16
8 / 16
1/2
Therefore, without using a calculator, the value of the expression "cos15 * 2sin15" is 1/2.