Find the exact value of the given expression using a sum or difference formula. sin 345°
Can you help with this? I got 1/4 (squareroot 2 - square root 6)
correct
Why not just check your result with a calculator?
I did but it isn't one of my answer options. The options are
a. (squareroot 3 - 1) / 2 squareroot 2
b. (squareroot 3 + 1) / 2 squareroot 2
c. (- squareroot 3 + 1) / 2 squareroot 2
d. (- squareroot 3 - 1) / 2 squareroot 2
poor choices for answers, they should have rationalized the answer.
That's where my hint about a calculator comes in.
Both our answers were (1/4)(√2 - √6)
if you work that out on your calculator you get
appr -.2588
working out c) you get appr -.2588
so c) is the correct choice, even though it does not look like ours
if you multiply c) by (√2/√2) , really just by 1, you get
((- √3 + 1) / 2√2 * (√2/√2)
= (-√6 + √2)/4
or
(1/4)(√2 - √6) , our answer
Certainly! To find the exact value of sin 345° using a sum or difference formula, we can start by recognizing that 345° can be expressed as the sum of two angles, namely 315° and 30°.
The sine of an angle can be expressed using the sum or difference formulas as:
sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
sin(A - B) = sin(A)cos(B) - cos(A)sin(B)
In this case, we can utilize the sum formula sin(A + B) = sin(A)cos(B) + cos(A)sin(B) with A = 315° and B = 30°.
sin 345° = sin(315° + 30°)
Using the sum formula:
sin(315° + 30°)
= sin(315°)cos(30°)+ cos(315°)sin(30°)
Now, we need to find the exact values of sine and cosine for 315° and 30°. Reference angles for these angles will be helpful in our calculations.
The reference angle for 315° can be found by subtracting 360° from it until we get a positive angle between 0° and 360°:
315° - 360° = -45° (out of the desired range)
315° - 360° = -45° (still out of the desired range)
315° - 360° = -45° (now we are in the desired range)
So, the reference angle for 315° is 45°.
Since sine is positive in the fourth quadrant, we can conclude:
sin 315° = sin 45°
Likewise, we can determine the reference angle for 30°:
30° - 360° = -330° (out of the desired range)
30° + 360° = 390° (out of the desired range)
30° + 360° = 390° (now we are in the desired range)
The reference angle for 30° is 30°.
Now we can substitute the exact values into the equation:
sin(315°)cos(30°) + cos(315°)sin(30°)
= sin(45°)cos(30°) + cos(45°)sin(30°)
Using the values for sine and cosine of 45° and 30°:
(sin(45°) * √2/2) * (√3/2) + (√2/2) * (1/2)
= √2/2 * √3/2 + √2/2 * 1/2
= (√6/4) + (√2/4)
= (√6 + √2) / 4
So, the exact value of sin 345° using a sum formula is (√6 + √2) / 4.
It seems that your answer of 1/4 (√2 - √6) is incorrect.