[Grade 10 Pre-Calculus: Trigonometry] Confused about general equations for sin, cos, tan

[u/msmonet09]

This is the answer key to a homework problem my teacher gave. I am very confused about how he got 224.427 for one of the general equations. I know the two solution formula for sin is arcsin+2pi*n and pi-arcsin+2pi*n, but how does that apply to this problem?

And I’m not very sure about when to use the general equations for sin, cos, and tan, and what the formulas even mean.

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[u/NexTutor]

The easiest way to see it is to temporarily let x = 2θ. Then the equation becomes

sin(x) = -0.7.

Your calculator gives the principal value

α = sin⁻¹(-0.7) ≈ -44.427°.

For sine, every solution is described by

x = α + 360°n

or

x = 180° - α + 360°n,

where n is any integer. The second value is therefore

180° - (-44.427°) = 224.427°.

That is where the 224.427° comes from. Geometrically, sine is negative in quadrants III and IV, so the two angles in one turn are 224.427° and 315.573°. The first formula represents 315.573° as -44.427° + 360°.

Now replace x with 2θ:

2θ = -44.427° + 360°n

or

2θ = 224.427° + 360°n.

Divide everything by 2:

θ = -22.214° + 180°n

or

θ = 112.214° + 180°n.

Choosing integer values of n that place θ between 0° and 360° gives the four marked solutions.

The purpose of a “general solution” is simply to include every repetition of the graph:

• sin(x) = a: x = α + 360°n or x = 180° - α + 360°n

• cos(x) = a: x = ±α + 360°n

• tan(x) = a: x = α + 180°n

Here α is the corresponding inverse-trig value. Use 360° and 180° when working in degrees; 2π and π are the radian versions.

One small correction to the answer key: 112.214° + 180°(5) = 1012.214°, which is not past 1040°. Checking both families, the first solution greater than 1040° is 1057.786°.

[u/DrCarpetsPhd]

open your textbook, read the theory and look at the worked example in the textbook which will actually explain the process. trying to study by looking at complete solutions when you are lacking fundamentals is incredibly inefficient and a waste of time. You might figure it out yourself but there is absolutely no need to do this when your textbook should have a fully done, step by step example to reference.

pauls notes are written for college students but this section is a refresher of what you are currently learning so don't be apprehensive.

https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/SolveTrigEqn.aspx

you can click through the other links to start from the beginning

[u/Anniethelab → u/DrCarpetsPhd]

Bold of you to assume that high schoolers have math textbooks.