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### Course: Trigonometry>Unit 4

Lesson 3: Sinusoidal models

# Interpreting solutions of trigonometric equations

Starting from a context represented by a trigonometric function, interpret equations based on the function. Created by Sal Khan.

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• I am a bit confused at 3.57, how is the bar at its lowest height when sine is at its max won't it be representing the maximum point the bar can go to since sin 1 is the highest point in the sine function
• No, the solution set 95 = 90 -12 * sin (pi/2) is representing the minimum value, not the maximum.

The equation is B(95)= 90 - 12 sin (pi/2) not 90 +12 *sin (pi/2), therefore it is going to be 90 -12 * 1 = 78.

When it is 90 -12*sin (3pi/2) *(opposite on the unit circle), it is 90 +12 = 112.

If the equation was 90 +12 *sin (pi/2) = 90 + 12 =112 and 90 +12 sin (3pi/2) = 90-12 = 78.

Just think that if the sign before sin/cos function determines what direction it will go to.

If it the sign in front of sin is positive, then when it is sin(pi/2) it will be the maximum. Ex: 12 + sin (pi/2) = 13 or -33 + 30*sin (pi/2) = -3. Also when it is sin (3pi/2) that will be the minimum: Ex: 12 + sin (3pi/2) = 11 or -33 + 30*sin (3pi/2) = -63

If the sign in front of sin in negative, then if it is sin(pi/2) it is the minimum and when it is sin(3pi/3 it is the maximum):

Ex: 41 - 13*sin (pi/2) = 41 -13 = 28
41 - 13*sin (3pi/2) = 41 + 13 = 54

Ex: 79 - 9*sin (pi/2) = 79 - 9 =70
79 - 9*sin (3pi/2) = 79 + 9 = 88

Check this link out to visualize the equation shown above on a graph: https://www.desmos.com/calculator/uqmquflq9m

As you can see, on y = 90 - 12*sin (5t), when x= pi/2, y = 78. On the other graph, y =90 +12*sin (5t), when x=pi/2, it is the maximum value with y =112.

I hope that I was able to explain it well enough to you. Do tell me if you have any questions.
• For the third question, I would have say let pi/2 = 5t, then solve for t, then the solution set is the height of the bar at pi/10 seconds.

But yours works too .. :)
• What does the least positive solution mean? For example, what is the least positive solution for 38 = 35 + 9cos(2.4t)? Also, what does it the least positive solution represent in this case?
• A good representation of this problem is thinking of it as a function:
f(x) = 10+9cos(2.4t)
We know that cos(x) is a cycle function, and by it's own f(x)=cos(x) has the highest solution of 1 (where x = 0) and lowest solution of -1 (where x = pi/2).

Therefore, to find the answer to the least positive solution, we just need to find the value where cos(2.4t) = 0. When cos(2.4t) = 0, notice that the only term we have left is 10: the lowest possible solution.
Maybe if this is a temperature function, the least positive solution would be the coldest temperature.
• Lol when his pointer went near the treadle it just disappeared :O
(1 vote)
• Conspiracies, conspiracies...
• In my Math career, I have the seen the amplitude refer to an absolute value (distance above or below the midline), but I have also seen it represented with a sign, indicating whether the sinusoidal wave goes up or down first.

So given some equation such as y = 3-4cos(2x + 5*PI), would the amplitude be 4 or -4?

Thanks!
(1 vote)
• The amplitude is always positive, so the amplitude here is 4. Whether the coefficient on the trig function is positive or negative/whether the wave starts by moving up or down is a different piece of information, separate from the amplitude.