Trigonometric Equations: Solve sin 3x - sin x = cos x, cos 2x + 3 sin x = 2

[Sam94]

Hi,

Could anyone kindly help me out with these following equations. I assume some of you here may find them straight forward and basic but I'm really struggling with this topic. I will appreciate any kind help. Thanks!

Solve in the range 0 ≤ θ ≤ 360

1. Sin 3θ - Sinθ = Cos θ
2. Cos 2θ + 3 Sinθ = 2

The following one is heavy and I just want to get an idea of how to tackle the second the task marked as b.

• Two voltages ?? = ?. ? ??? ? and ??= ?. ? ??? ? are inputs to an analogue circuit.

a. Use spreadsheet software to plot the two given voltages on the same pair of axis in the
range -???° = ? = ???°.

b. If the output voltage ???? is given by (??+ ??), use the plots in part (a) to determine an
expression for ???? in the form ? ???(? + ?).

c. Determine analytically the value of ? in the range 0° = ? = 360° when the output
voltage is 7.5 volts.


Many thanks to whoever gives some effort and helps me out.

 

[Dr.Peterson]

Solve in the range 0 ≤ θ ≤ 360

1. Sin 3θ - Sinθ = Cos θ
2. Cos 2θ + 3 Sinθ = 2

The following one is heavy and I just want to get an idea of how to tackle the second the task marked as b.

• Two voltages ?? = ?. ? ??? ? and ??= ?. ? ??? ? are inputs to an analogue circuit.

a. Use spreadsheet software to plot the two given voltages on the same pair of axis in the
range -???° = ? = ???°.

b. If the output voltage ???? is given by (??+ ??), use the plots in part (a) to determine an
expression for ???? in the form ? ???(? + ?).

c. Determine analytically the value of ? in the range 0° = ? = 360° when the output
voltage is 7.5 volts.

For the first two, the first step is to use the multiple-angle formulas to rewrite sin 3θ and cos 2θ in terms of sin θ and cos θ. Have you done that? You may not have been explicitly taught the triple-angle formulas, but you can create them using double angle formulas by way of sin(2θ + θ).

Do that, see if the result gains you anything, and then show what you got so we can correct it or make a next suggestion. I think the second should be easier, so try that first.

We can't help you with part (b) of the third problem without seeing your plots from part (a), since it depends greatly on that. It appears that you are supposed to plot the sum, then write an (approximate?) formula for it (using techniques you have presumably learned for doing so, by looking for peaks or crossings), and then perhaps check. So please do part (a), ideally including a plot of the sum, and send us a screen shot and tell us what you know about writing an equation for it.

 

[stapel]

Could anyone kindly help me out with these following equations.

Certainly! But first, we'll need to see what you've tried and where you're getting stuck, so we know how to help. So please reply with that information. For instance:

Solve in the range 0 ≤ θ ≤ 360

1. Sin 3θ - Sinθ = Cos θ

What identity (or identities) did you apply, to convert the sine of 3θ to sines and/or cosines of plain θ? Where did this lead?

2. Cos 2θ + 3 Sinθ = 2

You applied the double-angle identity to convert the cosine of 2θ to sines of plain θ. This gave you a quadratic in sin(θ). You factored in the usual manner, and... then what?

The following one is heavy and I just want to get an idea of how to tackle the second the task marked as b.

What is meant by "heavy"?

3. Two voltages ?? = ?. ? ??? ? and ??= ?. ? ??? ? are inputs to an analogue circuit.

a. Use spreadsheet software to plot the two given voltages on the same pair of axes in the range -???° = ? = ???°.

b. If the output voltage ???? is given by (??+ ??), use the plots in part (a) to determine an expression for ???? in the form ? ???(? + ?).

You're supposed to use what you got in part (a) for completing part (b). What did you get for part (a)? (Note: Determining the required function can be done via algebraic techniques, too.)

c. Determine analytically the value of ? in the range 0° = ? = 360° when the output voltage is 7.5 volts.

Once you determine the function in (b), plug "7.5" in for v_out, and solve for the angle measure.