Evaluating trig identities without calculator

[PCSL]

How do you do stuff like sin(pi/4) in your head or sin(1) or sec(pi(x))? Thanks for your help. I'm in Calc II and fully understand the calculus but my trig foundation from high school isn't the best.

 

[ArcanaNoir]

Some people use the unit circle, but I use special triangles and the graphs of sin, cosine and tangent. If you memorize the value at 0, and learn just a couple other key features of the graphs, that helps tremendously with negative values and positive values that exceed 2pi.

I use the 30-60-90 triangle and the 45-45-90 triangle. (refers to angle measures). Those triangles and the memorization of the graphs pretty much covers all angles you could expect to be asked to know without a calculator.

The sides of a 30-60-90 triangle are 1, 2, and $\sqrt{3}$.
The 45-45-90 is 1, 1, and $\sqrt{2}$.

 

[Mark44]

How do you do stuff like sin(pi/4) in your head or sin(1) or sec(pi(x))? Thanks for your help. I'm in Calc II and fully understand the calculus but my trig foundation from high school isn't the best.

There are a small angles whose sines and cosines you should have memorized:
Degrees Radians
0 0
30 pi/6
45 pi/4
60 pi/3
90 pi/2

From geometry, a 30 (deg) - 60 - 90 triangle has a hypotenuse that's twice as long as the short side, so sin(30 deg) = sin(pi/6) = 1/2 and cos(30 deg) = cos(pi/6) = sqrt(3)/2.

Looking at the other acute angle in this triangle, you can see that sin(60 deg) = cos(30) = sqrt(3)/2, and that cos(60 deg) = sin(30 deg) = 1/2.

A 45 (deg) - 45 - 90 right triangle has equal legs. If each leg is 1 unit, then by the Pythagorean Theorem, the hypotenuse = sqrt(1^2 + 1^2) = sqrt(2).
From this you can see that sin(45 deg) = sin(pi/4) = 1/sqrt(2), which is the same as sqrt(2)/2. Also, cos(45 deg) = cos(pi/4) = sqrt(2)/2.

Sine and cosine are usually presented using the unit circle, where one ray from an angle theta (measured counterclockwise with respect to the positive x-axis) intersects this circle at a point (x, y). The x-coordinate gives the cosine of theta; the y-coordinate give the sine of theta.

At the point (0, 1), which corresponds to and angle of 90 deg. or pi/2, cos(pi/2) = 0 and sin(pi/2) = 1.

Nobody is going to expect you to have sin(1) committed to memory. I'm not sure what you mean by sec(pi(x)). Do you mean sec(pi * x)? Depending on what x is, this might or might not be something that a teacher would expect you to calculate without a calculator.

 

[PCSL]

Thanks a lot! One more question - not really related to the forum I posted this in... is there any value in memorizing reduction formulas? My prof said he will provide the reduction formulas we need on the test but for upper level math classes I was wondering if having them memorized would help.

 

[Mark44]

Do you mean, formulas such as sin(2x) = 2sin(x)cos(x) and the like? If so, these are worth memorizing, IMO.

 

[QuarkCharmer]

Do you mean, formulas such as sin(2x) = 2sin(x)cos(x) and the like? If so, these are worth memorizing, IMO.

If this is what he means, then absolutely. These things will pop up over and over all throughout calculus.

 

[dynamicsolo]

The main things I tell students going on to calculus or physics courses from their pre-calculus course (or sequence) that they should remember from trigonometry are:

the definitions of the six trig ratios/functions;

sine and cosine for the list of angles (both degrees and radians) that Mark44 shows; you can calculate the other four trig ratios for those angles from there; you should also know how to place the 30-60-90 and 45-45-90 triangles into the other quadrants, so you can get the exact trig values for all the multiples of 30º (or pi/6) and 45º (or pi/4) in the principal circle;

the Pythagorean Identity in its basic and two alternate forms: you use $\sin^{2} \theta + \cos^{2} \theta = 1$ a lot , $\tan^{2} \theta + 1 = sec^{2}\theta$ somewhat less often, and $\cot^{2} \theta + 1 = csc^{2}\theta$ not so much (but you shouldn't just fuggetaboudit...);

the "double-angle formulas" for $\sin( 2 \theta )$ and for $\cos( 2 \theta )$ in its basic and two alternate forms; you need the forms of "cosine double-angle" to get the rather handy substitutions for $\sin^{2} \theta$ and $\cos^{2} \theta$ .

Beyond that, you should know about the "angle-addition" formulas for sine, cosine, and tangent, the "half-angle formulas", the "product-to-sum formulas", and the "sum-to-product formulas". They're useful, but most people don't work with them so often that they'll stick in their heads well enough to keep them straight. Usually, practically everyone looks them up (but you have to know they exist to know there is something to look up!).

For applications, you should remember basically how to use the trig ratios to solve for sides or angles of right triangles. The Law of Sines and Law of Cosines are also important to remember if you're going to be doing applications much.

I think that pretty much covers memorable basic trig... Did I miss anything?Oh yeah, the ranges of the inverse trig functions, because that's what your calculator is using when it displays an answer; you should be aware that the answer The Machine gives you may be in the wrong quadrant for your application, so from there, you need to know how to adapt the displayed angle to the correct angle for your problem, as necessary...