On this page we hope to clear up problems you might have
with the trigonometric ratios. The trigonometric ratios
are very useful when dealing with triangles and unit circles.
Click any of the links below or scroll down to better your
understanding of the trigonometric ratios.
Reciprocal ratios (csc, sec, cot)
Rotations (unit circle)
Graphs involving the trig. ratios
Pythagorean and quotient identities
Quiz on Trigonometry
The trig. ratios, sine, cosine, and tangent are
based on properties of right triangles. The function values
depend on the measure of the angle. The functions are outlined
The reciprocal ratios are trigonometric ratios, too.
They are outlined below.
Angles are also called rotations because they
can be formed by rotating a ray around the origin on the coordinate plane.
The initial side is the x-axis and the ray that has
been rotated to form an angle is the terminal side.
1. Problem: Find the reference angle for theta (see the figure below).
Solution: To find the measure of the acute angle formed by the terminal side and the x-axis subtract the measure of theta from 180o. 180 - 115 = 65 The reference angle is 65o.Once you have found the reference angle, use it to determine the trig. function values. Consider, for example, an angle of 150o. The terminal side makes a 30o angle with the x-axis, since 180 - 150 = 30. As the figure below shows, triangle ONR is congruent to triangle ON'R'; therefore, the ratios of the sides of the two triangles are the same, although the ratios may have different signs. (You could determine the function values directly from triangle ONR, but that is not necessary if you remember that the sine is positive and the cosine and tangent are negative in quadrant II.)
Up until now, you have probably only measured angles using degrees.
Another useful measure, based on the unit circle, is called radians.
Radians scare intermediate algebra and AP Calculus students alike,
so don't get too worried if they seem complicated or useless to you.
1. Problem: Convert 60o to radians. Solution: Multiply 60o by (PI)/180o. 60o (PI) --- * ---- 1 180o 60o(PI) ------- 180o Perform the indicated division. Cancel out the degrees. (PI) ---- 3 2. Problem: Convert (3(PI))/4 to degrees. Solution: Multiply (3(PI))/4 by 180o/(PI). 3(PI) 180o ----- * ---- 4 (PI) 3(PI)180o --------- 4(PI) Perform the indicated division. ((PI) cancels out.) 3 - * 180o 4 135o
In a right triangle, the two acute angles are complementary. Thus, if one
acute angle of a right triangle is x, the other is 90o - x.
Therefore, if sin x = (a/c) then cos (90o - x) = (a/c).
A table of all the cofunctions is displayed below.
1. Problem: Find the function value of cot 60o. Solution: Use the cotangent's cofunction identity to rewrite the problem. tan (90o - 60o) tan 30o The tangent of 30o is one you should have memorized. (SQRT(3))/3
All six of the trigonometric functions are periodic, that is, their graphs
repeat after a certain period. The periods of the six trig. functions
are shown below.
There are two quotient identities. They tell us that the tangent and cotangent functions can be expressed in terms of the sine and cosine functions. They are listed below.
sin x tan x = -----, cos x <> 0 cos x cos x cot x = -----, sin x <> 0 sin x
There are three other identities that are very important. They are called
the Pythagorean Identities. The Pythagorean Identities come in handy
later on when you need to prove more complicated trig. identities equal.
The Pythagorean Identities are listed below.
Trigonometric expression such as tan(x - (PI)) represent numbers, just as algebraic expressions represent numbers. Since that is true, we know we can manipulate trig. expressions the same way we do algebraic expressions. Examples:
1. Problem: Simplify cos y(tan y - sec y). Solution: Use the distributive property of multiplication, which says a(b + c) = ab + ac. cos y(tan y) - cos y(sec y) Simplify the expression by writing it in terms of cos. Use the Quotient Identities. sin y 1 (cos y)----- - (cos y)----- cos y cos y Perform the indicated multiplications. (cos y)sin y cos y ------------ - ----- cos y cos y Perform the indicated divisions. sin y - 1 2. Problem: Simplify (sin2 x)(cos2 x) + cos4 x Solution: Factor. (cos2 x)(sin2 x + cos2 x) Using a Pythagorean Identity, replace sin2 x + cos2 x with 1. (cos2 x)(1) cos2 x
Take the Quiz on trigonometry. (Very useful to review or to see if you've really got this topic down.) Do it!