Direct quotes from the Trigonometry tutorial at David@catcode.com.
Here’s a basic right triangle. Remember, the line segment opposite the right angle in a right triangle is called the hypotenuse. Since Trig is all about the relationships of the sides and angles of triangles, let’s “do some Trig” and look at the angle A. The line segment opposite the angle A is called ??? The line segment closest to the angle A is called ??? So, we’ve given this triangle’s sides differing names. What about if we chose another angle?
Here we have the exact same triangle, but this time we’re trying to find out relationships between the sides and the angle B. Once again, the side opposite the angle B is called ??? And, the side closest to angle B is called ??? That tells us that the angle determines which side is the “opposite” side and which side is the “adjacent” side. This information is used in Trig to determine sine, cosine, tangent, cosecant, secant and cotangent.
These six trigonometric functions are the 6 different ratios that you can set up from a right triangle. (per PowerPoint presentation by Sally Keely, Trigonometric Functions Defined)
Here’s an easy way to remember the “formulas” for the three primary trig functions. If you learn these three and remember that cosecant is the reciprocal of sine, secant is the reciprocal of cosine and cotangent is the reciprocal of tangent, you’ll know all six trig functions!
How can we apply our knowledge of right triangles to the Cartesian coordinate system? In Trig, we use Greek letters as general terms to stand for the measures of angles. I used the Greek letter, theta, in this diagram. Can you see that the angle, theta, is drawn in standard position on the x-axis and terminates at the point (x,y)? Since Trig is the study of triangles, how could we make this angle into a right triangle? A line segment perpendicular to the x-axis drawn from point (x,y ) ending at the x-axis could serve as the second side of the triangle. Let’s name this side y. A line segment on the x-axis from the point of origination to the line segment y could be the third side of the triangle. We’ll call this side x; thereby, creating a right triangle! Let’s call the hypotenuse of the triangle r. If we knew the length of line segment x and the length of line segment y, how could we compute the length of line segment r in our right triangle? Who remembers the Pythagorean Theorem? X squared plus y squared equals r squared. If we knew the length of line segment x and line segment y, we could compute r by utilizing the Pythagorean theorem.
Here is a different way of looking at the six trigonometric functions. It’s the same as SOH/CAH/TOA, but applied to the x-y axis. These definitions need to memorized, but do you see some similarities? Sin and cos both have r as their denominator. Sin shows the relationship of y to r and cos shows the relationship of x to r. Tan shows the relationship of y to x. What about Csc, Sec and Cot? Do you see that Csc is, again , the reciprocal of sin? Sec is, again , the reciprocal of cos? And, Cot is, again , the reciprocal of tangent?
Now, what if we applied the definitions of the trig functions to this most basic circle.
Graphic from http://home.alltel.net/okrebs/page72.html Can you see that if you placed all triangles with the same r value, or radius, and differing angle values from 1 degree to 360 degrees you would get a circle if you connected all the “x,y” points? “We could generalize this to say that the circle of radius r describes the collection of all triangles with hypotenuse r . . . Any right triangle is similar to some right triangle with hypotenuse length 1.” (per http://mathforum.org/library/drmath/view/53944.html) In making the radius equal to 1, you’ve created what’s known as the unit circle.
How can we apply our knowledge of right triangles to the Cartesian coordinate system? In Trig, we use Greek letters as general terms to stand for the measures of angles. I used the Greek letter, theta, in this diagram. Can you see that the angle, theta, is drawn in standard position on the x-axis and terminates at the point (x,y)? Since Trig is the study of triangles, how could we make this angle into a right triangle? A line segment perpendicular to the x-axis drawn from point (x,y ) ending at the x-axis could serve as the second side of the triangle. Let’s name this side y. A line segment on the x-axis from the point of origination to the line segment y could be the third side of the triangle. We’ll call this side x; thereby, creating a right triangle! Let’s call the hypotenuse of the triangle r. If we knew the length of line segment x and the length of line segment y, how could we compute the length of line segment r in our right triangle? Who remembers the Pythagorean Theorem? X squared plus y squared equals r squared. If we knew the length of line segment x and line segment y, we could compute r by utilizing the Pythagorean theorem.
How can we apply our knowledge of right triangles to the Cartesian coordinate system? In Trig, we use Greek letters as general terms to stand for the measures of angles. I used the Greek letter, theta, in this diagram. Can you see that the angle, theta, is drawn in standard position on the x-axis and terminates at the point (x,y)? Since Trig is the study of triangles, how could we make this angle into a right triangle? A line segment perpendicular to the x-axis drawn from point (x,y ) ending at the x-axis could serve as the second side of the triangle. Let’s name this side y. A line segment on the x-axis from the point of origination to the line segment y could be the third side of the triangle. We’ll call this side x; thereby, creating a right triangle! Let’s call the hypotenuse of the triangle r. If we knew the length of line segment x and the length of line segment y, how could we compute the length of line segment r in our right triangle? Who remembers the Pythagorean Theorem? X squared plus y squared equals r squared. If we knew the length of line segment x and line segment y, we could compute r by utilizing the Pythagorean theorem.
How can we apply our knowledge of right triangles to the Cartesian coordinate system? In Trig, we use Greek letters as general terms to stand for the measures of angles. I used the Greek letter, theta, in this diagram. Can you see that the angle, theta, is drawn in standard position on the x-axis and terminates at the point (x,y)? Since Trig is the study of triangles, how could we make this angle into a right triangle? A line segment perpendicular to the x-axis drawn from point (x,y ) ending at the x-axis could serve as the second side of the triangle. Let’s name this side y. A line segment on the x-axis from the point of origination to the line segment y could be the third side of the triangle. We’ll call this side x; thereby, creating a right triangle! Let’s call the hypotenuse of the triangle r. If we knew the length of line segment x and the length of line segment y, how could we compute the length of line segment r in our right triangle? Who remembers the Pythagorean Theorem? X squared plus y squared equals r squared. If we knew the length of line segment x and line segment y, we could compute r by utilizing the Pythagorean theorem.
There are many special angles in the unit circle that we will need to know the sin and cos values for. We will need to know the sin, cos, tan, etc. for 30 degrees, 45 degrees, 60 degrees, 90 degrees, etc.. Remembering that y equals sin on the unit circle and x equals cos, this diagram gives us the sin and cos for numerous special angles. The exact values for these special angles must, also, be committed to memory.
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