Hi, I need help with this question:
Use the addition and subtraction formulas ti fubd the exact values of the trigonometric ratios:
cos(-7pi/12)
Okay, I'm just having trouble with a rather basic part of the question...
I wrote cos(pi/4 + pi/3), but now I realized that the trig ratio is negative. How do I write -7pi/12 as the sum/difference of two trig ratios?
To write -7pi/12 as the sum or difference of two trigonometric ratios, you need to find two angles whose sum or difference is -7pi/12.
One way to do this is to start with an angle that is close to -7pi/12, such as -3pi/4, which is equivalent to -9pi/12. Then, you can subtract pi/12 from -3pi/4 to get -7pi/12:
-3pi/4 - pi/12 = -9pi/12 - pi/12 = -10pi/12 = -5pi/6
So, you can write -7pi/12 as the difference of -3pi/4 and pi/12:
-7pi/12 = -3pi/4 - pi/12
Now you can use the subtraction formula for cosine, which states that:
cos(a - b) = cos(a)cos(b) + sin(a)sin(b)
In this case, you can substitute -3pi/4 for a and pi/12 for b:
cos(-7pi/12) = cos(-3pi/4 - pi/12)
To find the exact value of cos(-7pi/12), you'll need to use the values of cos(-3pi/4) and cos(pi/12), which can be found using the unit circle or a calculator.
To express -7π/12 as the sum or difference of two trigonometric ratios, you can use the fact that π is equivalent to 12π/12.
First, find a common denominator for π/4 and π/3.
The common denominator for both fractions is 12. Rewriting π/4 and π/3 using this denominator, we have:
π/4 = (3π/12)
π/3 = (4π/12)
Now, we can rewrite -7π/12 as:
-7π/12 = -((3π/12) + (4π/12))
Simplifying this expression, we have:
-7π/12 = -(7π/12)
So, the exact value of cos(-7π/12) is the same as cos(7π/12), but with a negative sign.