How NOT to Memorise Exact Values!In my recent thread
Introduction to Trigonometry!, I explained basics of the unit circle including definitions of sin, cos and tan in terms of my own understandings. In this thread I will be explaining how not to memorise the exact values, but actually understand
why they work, which will ultimately help with memorising them, so you're still memorising (but with a bit more reasoning)!
So just to revisit, in the image below you can see that at an angle of
from the positive x-axis, sin is defined as the
y-coordinate (blue arrow) and cos is defined as the
x-coordinate (red arrow). Have a close look at point P whilst looking at the images below to help you understand what is going on.
Exact valuesWhat are they?The exact values of trigonometric functions are the
exact values of sin, cos and tan at the 5 angles that are within the first quadrant. Many people memorise these values, and that's great - not
that easy to memorise 5 random values! - but there are actual patterns and intuitive meanings towards the exact values. When you learn these patterns, you will have memorised the exact values easily, and will probably not forget any of them, as you will know exactly
why they are!
So, we start all our angles off the positive x-axis (see my introduction to trig for further details). Before we even move any angle, we remain at the angle of 0, where there is no
y-value (sin - blue arrow), but there is an
x-value (cos - red arrow). We also know that if the point is touching the circumference, the radius is 1, despite where we are. So at cos(0), the x-value is 1. But what is sin(0), the y-value at zero degrees? You tell me, is there a blue arrow on the first image? No? Exactly, because at zero degrees, we have not moved up yet, which means the y-values have not started, hence sin(0) = 0. Hopefully you are on the same page as me (if not, below there are interactive links in which you can experiment), now as the angle theta becomes larger, you can see that
cos (red arrow) is getting smaller, whereas
sin (blue arrow) is getting larger. When we reach 90 degrees, there is another distinctive feature. Sin(90) describes the y-value at 90 degrees, which is 1 as it's clearly the radius. What about cos(90)? What is the x-value at 90 degrees? Zero? Exactly, there is no x-axis at 90 degrees, hence cos(90) = 0.
The first quadrant is broken up into 5 angles: 0, 30, 45, 60 and 90. Now as mentioned above, the x-values get smaller as theta (the angle) gets bigger. So cos(0) = 1, cos(30) = some value smaller than 1, cos(45) = some value smaller than cos(30) and so on. If this is the case, then as theta gets larger, so does sin. These are our exact values, where I will give a brief description on below:
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As you can see above, the values increase from 0 to 1. So now that we know the exact values of these figures above, we can apply them to sin and cos. We said that cos(0) = 1 and as the angle increases, the value of cos decreases too. So if we apply the pattern that cos is decreasing as the angle is increasing,
cos(30) = 0.866 (fourth smallest value),
cos(45) = 0.707 (third smallest value),
cos (60) = 0.5 (second smallest value) and
cos(90) = 0 (smallest value). Lets take a look at sin. We said that sin increases as the angle gets larger. So if
sin(0) = 0,
sin(30) = 0.5 (fourth largest value),
sin(45) = 0.707 (third largest value),
sin(60) = 0.866 (second largest value) and finally
sin(90) = 1 (largest value).
Helpful links to help you visualise exact values:
https://www.mathsisfun.com/algebra/trig-interactive-unit-circle.htmlhttp://www.mathmistakes.info/facts/TrigFacts/learn/vals/sum.htmlhttps://www.youtube.com/watch?v=NDct0Se3Q3khttps://www.youtube.com/watch?v=izxNp8FsPM8https://www.youtube.com/watch?v=NO4H4YROdqkThere we have it! Hopefully that made sense to you all, and if it didn't do not be disheartened, it may take a while to understand (sure took me a while haha), but you will eventually understand it, I guarantee you. If there are any suggestions, questions or issues please PM me, or leave a comment here. Enjoy and good luck!
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