Compound Angles: Find Exact Value of cos(a+b)

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In summary: For example, for angle a, in the given circle, you know that the sine of angle a is 2/5. What does that tell you about the ratio of the opposite side to the hypotenuse?In summary, angle a is located in the second quadrant with a sine value of 2/5 and angle b is located in the third quadrant with a tangent value of 3/4. To find the exact value of cos(a+b), we can use the formula cos(a+b)=cosacosb-sinasinb and solve for cos(a) and sin(b) using right triangle trigonometry. This will give us the exact value of cos(a+b) and it will be located in the second or third quadrant
  • #1
xcortz
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Homework Statement


Angle a is located in the second quadrant where sin a=2/5 and angle b is located in the third quadrant where tan b=3/4. Determine an exact value for cos(a+b) and where it is located.

Homework Equations


cos(a+b)=cosacosb+sinasinb

The Attempt at a Solution


I don't know what I'm doing
 
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  • #2
xcortz said:

Homework Statement


Angle a is located in the second quadrant where sin a=2/5 and angle b is located in the third quadrant where tan b=3/4. Determine an exact value for cos(a+b) and where it is located.

Homework Equations


cos(a+b)=cosacosb+sinasinb

The Attempt at a Solution


I don't know what I'm doing

Check your formula in (2.) carefully.
 
  • #3
Ray Vickson said:
Check your formula in (2.) carefully.
cos(a+b)=cosacosb-sinasinb
 
  • #4
xcortz said:
cos(a+b)=cosacosb-sinasinb

OK, so what is preventing you from applying that?
 
  • #5
Ray Vickson said:
OK, so what is preventing you from applying that?
They want an exact value, not a decimal answer. I don't know what the exact value is for sin2/5 or tan3/4
 
  • #6
xcortz said:
They want an exact value, not a decimal answer. I don't know what the exact value is for sin2/5 or tan3/4

Why do you think you need to know those values? Read the question again, carefully.
 
  • #7
.
 
  • #8
Ray Vickson said:
Why do you think you need to know those values? Read the question again, carefully.
Because when I plugged the numbers in and solved for an exact value, I got 12.32/5 and it was wrong apparently
 
  • #9
xcortz said:
Because when I plugged the numbers in and solved for an exact value, I got 12.32/5 and it was wrong apparently

Let me repeat: read the question carefully, and show your work. We cannot tell where you are going wrong if we cannot see the details of what you have done.
 
  • #10
xcortz said:
They want an exact value, not a decimal answer. I don't know what the exact value is for sin2/5 or tan3/4
These value are not correct. It's not sin(2/5) -- it's sin(a) = 2/5, and similar for angle b. You are given that sin(a) = 2/5. Using right triangle trig, you should be able to find the exact value for cos(a) (don't use a calculator), and similarly for sin(b) and cos(b).
 
  • #11
Think about what sine represents in terms of sides of triangle.
 

1. What is a compound angle?

A compound angle is an angle that is formed by adding or subtracting two or more individual angles. It is denoted as (a+b) or (a-b), where a and b are the individual angles.

2. How do you find the exact value of cos(a+b)?

To find the exact value of cos(a+b), you can use the double angle formula for cosine, which is cos(a+b) = cos(a)cos(b) - sin(a)sin(b). You will need to know the exact values of cos(a) and sin(a) to substitute into the formula.

3. Can you use a calculator to find the exact value of cos(a+b)?

No, a calculator can only give you an approximate value for cos(a+b). To find the exact value, you will need to use mathematical formulas and the exact values of cos(a) and sin(a).

4. What is the difference between a compound angle and a simple angle?

A simple angle is an angle that is not formed by adding or subtracting other angles. It is just one single angle. A compound angle, on the other hand, is formed by adding or subtracting two or more individual angles.

5. Why is it important to find the exact value of cos(a+b)?

Finding the exact value of cos(a+b) is important because it allows us to solve more complex trigonometric equations and better understand the relationship between different angles. It also helps us to make precise calculations in fields such as engineering, physics, and astronomy.

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