Evaluating Trig Functions at Special Right Triangles
(With Mr. Soh Cah Toa!)
Hello everyone! I'm Mr. Soh Cah Toa, and I'm here today to help you all learn to evaluate trig functions at special right triangles. My name, Soh Cah Toa, is a useful tool to help you remember trig ratios. If needed, reference the link down below for all of the ratios. Now, let's begin!
Me again! First, I'll introduce you to our two special right triangles, Mr. 45°-45°-90° and Ms. 30°-60°-90°! Every triangle has 180°. No matter how big these triangles are, they will always have the same angle measures. With that said, the ratio of their sides will always be the same! How reliable!
Mr. 45°-45°-90°*Keep in mind that cosecant is the reciprocal of sine, secant is the reciprocal of cosine, and cotangent is the reciprocal of tangent.
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Ms. 30°-60°-90° *Keep in mind that cosecant is the reciprocal of sine, secant is the reciprocal of cosine, and cotangent is the reciprocal of tangent.
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Ha ha ha, they're a great couple aren't they. And, how fun was that!? Moving on now... I will tell you about reference angles! Every angle (with the exception of quadrantal angles of course) has a reference angle. This angle is an acute angle between the terminal side of an angle and the x-axis. Take a look by clicking on "reference angles" above.
Using reference angles, you can evaluate trig functions too! When evaluating trig functions for an angle, use the referance angle to determine the trig ratios. Let's look at an example to show you exactly what I mean here!
Example One
This is a 135 degree angle. With a 45 degree reference angle. When solving for the six trig functions, use the 45 degree reference angle which will give you a 45°-45°-90° triangle.
Pay attention to negatives and positives. Since this triangle is on the negative side of the x-axis, the "adjacent" side is negative. This will affect cosine, secant, tangent, and cotangent.
sin(135°) = 1/√2 or √2/2
cos(135°) = -1/√2 or -√2/2
tan(135°) = -1/1 or -1
csc(135°) = √2/1 or √2
sec(135°) = -√2/1 or -√2
cot(135°) = -1/1 or -1
Pay attention to negatives and positives. Since this triangle is on the negative side of the x-axis, the "adjacent" side is negative. This will affect cosine, secant, tangent, and cotangent.
sin(135°) = 1/√2 or √2/2
cos(135°) = -1/√2 or -√2/2
tan(135°) = -1/1 or -1
csc(135°) = √2/1 or √2
sec(135°) = -√2/1 or -√2
cot(135°) = -1/1 or -1