Is My Solution for 4sinx - 3cosx Correct?

In summary, the conversation involves someone unsure about their answer to 4sinx - 3cosx. They provide their answer of 5sin(x + sin^-1 (-3/5)) and mention using equations involving asinx + bcosx and angle addition identity to solve it. They request clarification and it is suggested to check their answer using the angle addition identity and facts involving arcsin and arccos. The person then confirms that their answer is correct and the equation has been fully solved.
  • #1
106
0
I'm unsure as to whether I'm correct with this:

4sinx - 3cosx

My answer ended up being 5sin(x + sin^-1 (-3/5)).

The equations I used were:

asinx + bcosx = (a^2 + b^2)^1/2 * sin(x + angle)
angle = sin^-1 (b / (a^2 + b^2)^1/2))

Can someone clarify whether I'm correct? I would appreciate it.
 
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  • #2
A good way to check your answer is to reverse the process using the angle addition identity:

sin (θ + φ) = sin θ cos φ + cos θ sin φ

And using the facts

sin arcsin (3/5) = 3/5
and
cos arcsin (3/5) = 4/5. (why?)
 
  • #3
I checked my answer and it equals the original equation, but have I fully solved it?
 
  • #4
If it checks (and I agree that it does), then yep!
 

What are sine and cosine?

Sine and cosine are two mathematical functions that are commonly used in trigonometry. They are ratios of the sides of a right triangle and are used to solve problems involving angles and distances.

What are some real-world applications of sine and cosine?

Sine and cosine have many practical applications, such as in navigation, engineering, and physics. They are used to calculate the height of buildings, the distance between two points, and the trajectory of a projectile.

How do you solve sine and cosine problems?

To solve sine and cosine problems, you will need to use the values of the given angles and sides of a triangle. You can then use the trigonometric ratios of sine and cosine to find missing values.

What is the difference between sine and cosine?

The main difference between sine and cosine is the ratio they represent. Sine is the ratio of the opposite side to the hypotenuse, while cosine is the ratio of the adjacent side to the hypotenuse.

Can you use sine and cosine for non-right triangles?

No, sine and cosine can only be used for right triangles, as they are defined based on the sides of a right triangle. Other trigonometric functions, such as tangent and cotangent, can be used for non-right triangles.

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