- #1
sunnybrooke
- 19
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Homework Statement
Homework Equations
The Attempt at a Solution
I can solve the first part of the question. -2 ≤ k ≤ 2 because -1 ≤ sin(x) ≤ 1. How do I solve the second part of the question? Thanks.
Michael Redei said:Forget k for a moment, and think about what values 2 sin(3x) can assume. For each such value v, you have 2 sin(3x) = v, or 2 sin(3x) + (-v) = 0. So your k values are the same as the values of -v.
sunnybrooke said:http://www4c.wolframalpha.com/Calculate/MSP/MSP4251a45574034148cah00001164cg2h50a94c78?MSPStoreType=image/gif&s=5&w=300&h=183&cdf=RangeControl
So if k:
is = -2, there will be 2 zeros
is between (-2,0] there will be 4 zeros
is between (0,2), there will be 2 zeros
is = 2, there will be 1 zero
Is that correct? Thanks.
sunnybrooke said:http://www4c.wolframalpha.com/Calculate/MSP/MSP4251a45574034148cah00001164cg2h50a94c78?MSPStoreType=image/gif&s=5&w=300&h=183&cdf=RangeControl
So if k:
is = -2, there will be 2 zeros
is between (-2,0] there will be 4 zeros
is between (0,2), there will be 2 zeros
is = 2, there will be 1 zero
Is that correct? Thanks.
piercebeatz said:Very close. However, the only time that there are 4 solutions is when k=0. For k (-2,0), there are 3 solutions
Michael Redei said:That looks correct to me, but you should check your result by looking at a graph of the function, perhaps with k=0 if you haven't already done so.
One other small matter, which you probably take for granted, but you should still mention if this is a homework or exam problem: what happens when k > 2 or k < -2?
A trig equation with 2 variables is an equation that contains both a trigonometric function (such as sine, cosine, or tangent) and two unknown variables (typically x and y).
To solve a trig equation with 2 variables, you need to use trigonometric identities and algebraic techniques to isolate one variable on one side of the equation, and then use substitution to solve for the other variable.
Some common strategies for solving trig equations with 2 variables include using the Pythagorean identity, using the double angle formula, and using the sum and difference formulas.
Yes, a trig equation with 2 variables can have multiple solutions. This is because trigonometric functions are periodic, meaning they repeat their values at regular intervals. Therefore, an equation may have multiple solutions that satisfy the given conditions.
Solving trig equations with 2 variables can be useful in fields such as engineering, physics, and astronomy. For example, in engineering, trigonometric equations can be used to calculate angles and distances in structural design. In physics, trigonometric equations can be used to model the motion of objects in 2 dimensions. In astronomy, trigonometric equations can be used to calculate the positions of celestial bodies in the sky.