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## Algebra II (2018 edition)

### Course: Algebra II (2018 edition)>Unit 9

Lesson 11: Constructing sinusoidal functions

# Trig word problem: modeling annual temperature

Sal solves a word problem about the annual change in temperature by modeling it with a sinusoidal function. Created by Sal Khan.

## Want to join the conversation?

• At , there is a mention of solving the problem in next video. But the next video is about modeling with phase shifts. Can anyone please point me to the solution video?
• Why is Sal using a whole bunch of terminology and concepts that we haven't learned yet?
• Many of these concepts aren't introduced until the next section "GRAPHS OF TRIG FUNCTIONS"
• What does Sal mean by argument?
• The argument is the input into a function. For example sin(x) has the argument x.
• How did the trigonometric functions be suddenly associated with data graphs? Previous lessons have only been showing angles in real life, like the sun to the ground, or a person to a tree, or a tower to the shadow? This video feels out of place. How do I know when to use a trigonemtric function to interpret data?
• You are thinking about "right triangle trigonometry", relating the sides of a right triangle. This extends to "circular trigonometry", relating the horizontal and vertical coordinates of a circular motion. Any situation that is based on circular motion may be modeled by a sinusoid, including the phases of the moon, day length as a function of the season, or tides.

As Noble notes, the sine and cosine also describe periodic motion where the acceleration is proportional and opposite to the position -- including EM waves, sound waves, and coil springs.
• How can we choose which function (sine / cosine / tangent) is used in a situation?
• If we are studying a periodic function, we would pay attention to the shape of the curve. Tangent can be separated out immediately because it is not this sinuous wave form. It is more like a flight of bats (curves swooping up from negative infinity toward positive infinity every so often, or swooping downward, if the function is negative). Sine waves of y= sin (x) pass through the value of 0 when the angle is 0. For a simple sine wave, the highest point of the snake will be 1 and the lowest point will be -1. After a little practice, cosine is easily identifiable because y = cos (x) passes through a value of 1 when the angle is 0. the highest point of the curve for y = cos (x) is also plus 1 and the lowest point is also -1. If the curves are happening more frequently or the waves are deeper (the variation between lowest and highest point), then the function may be more complicated, such as y = 5 sin (ᴨ/3) -12
With some practice and a few more video tutorials, it will become much easier to figure out a starting point and how to modify the basic shapes to fit the situation--wave heights pushing a toy boat, temperature variation, length of daylight or whatever.
• Where can I find the second part of the video?
• Hello Sal! Would the equation not be 7.5cos[ 2pi/365(x-7)] + 21.5? It states that the hottest day of the year (aka the maximum point, aka the START of the cosine function) is january 7th so would you not have to shift the entire graph to the right 7 days?
• If we set January 7th as the point d=7, then yes. In this case we set January 7th as d=0 though.
• "Because we're using a trig function so we're gonna hit our low-point exactly in between " . What? Is this a general rule?
• All Trig functions are periodic, so their minimums and maximums will be predictable since they'll just repeat again and again as x--> infinity or - infinity.

For example: y = sin(x)

This function has a repeating maximum at y = 0 and y = 1.

Hope that helps!
(1 vote)
• At 6.54, Sal mentioned that we will answer the question on how many days after Jan 7 is the first Spring day when temperature reaches 20 degrees. I could not find the next video.