Algebra II: Trigonometry

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. Read on or follow any of the links below to better your understanding of the trigonometric ratios.

Ratios (sin, cos, tan)
Reciprocal ratios (csc, sec, cot)
Rotations (unit circle)
Radians
Cofunctions
Graphs involving the trig. ratios
Pythagorean and quotient identities
Algebraic manipulation
Quiz on trigonometry

Ratios

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 below.

sine x = (side opposite x)/hypotenuse

cosine x = (side adjacent x)/hypotenuse

tangent x = (side opposite x)/(side adjacent x)

In the accompanying figure, sin A = a/c, cosine A = b/c, and tangent A = a/b.

There are two special triangles you need to know — 45-45-90 and 30-60-90 triangles. They are depicted in this accompanying figure and this accompanying figure, respectively.

The figures show how to find the side length of those types of triangles. Besides knowing how to find the length of any given side of the special triangles, you need to know their trig. ratio values (they are alwyas the same, no matter the size of the triangle because the trig. ratios depend on the measure of the angle). A table of these values is given in a figure.

(If you need a more in-depth understanding of the trig. ratios, you can follow this link and go to a geometry lesson that goes into great detail concerning the trigonometric ratios.)

Reciprocal Ratios

The reciprocal ratios are trigonometric ratios, too. They are outlined below.

cotangent x = 1/tan x = (adjacent side)/(opposite side)

secant x = 1/cos x = (hypotenuse)/(adjacent side)

cosecant x = 1/sin x = (hypotenuse)/(opposite side)

Rotations of Angles

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.

Example: Example figure.

Reference angles are useful when dealing with rotations that end in the second, third, or fourth quadrants. A reference angle for a rotation is the acute angle formed by the terminal side and the x-axis.

Example:

1.  Problem: Find the reference angle for theta (see the
             accompanying figure).
   Solution: To find the measure of the acute angle formed by the terminal
             side and the x-axis, subtract the measure of theta from
             180°.

             180 - 115 = 65

             The reference angle is 65°.

Once you have found the reference angle, use it to determine the trig. function values. Consider, for example, an angle of 150°. The terminal side makes a 30° angle with the x-axis, since 180 - 150 = 30. As the accompanying figure 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.)

Radians

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.

The accompanying figure shows measures in degrees and radians on the unit circle that you should probably memorize, as they are commonly used measures.

Sometimes it will be necessary to convert from radians to degrees or vice versa. To convert from degrees to radians, multiply by ((PI)/180°). To convert from radians to degrees, multiply by (180°/(PI)).

Examples:

1.  Problem: Convert 60° to radians.
   Solution: Multiply 60° by (PI)/180°.

             60°   (PI)
             --- * ----
              1    180°

             60°(PI)
             -------
              180°

             Perform the indicated division.  Cancel out the degrees.

             (PI)
             ----
               2 

2.  Problem: Convert (3(PI))/4 to degrees.
   Solution: Multiply (3(PI))/4 by 180°/(PI).

             3(PI)   180°
             ----- * ----
               4     (PI)

             3(PI)180°
             ---------
               4(PI)  

             Perform the indicated division.  ((PI) cancels out.)

             3
             - * 180°
             4

             135°

Cofunctions

In a right triangle, the two acute angles are complementary. Thus, if one acute angle of a right triangle is x, the other is 90° - x. Therefore, if sin x = (a/c) then cos (90° - x) = (a/c). A table of all the cofunctions is displayed below.

sin x = cos (90° - x)

tan x = cot (90° - x)

sec x = csc (90° - x)

cos x = sin (90° - x)

cot x = tan (90° - x)

csc x = sec (90° - x)

Example:

1.  Problem: Find the function value of cot 60°.
   Solution: Use the cotangent's cofunction identity to rewrite the 
             problem.

             tan (90° - 60°)
             tan 30°

             The tangent of 30° is one you should have memorized.

             (SQRT(3))/3

Graphing the Trigonometric Ratios

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.

sin, cos, csc, and sec = 2(PI)

tan and cot = PI

The sin and cos graphs have a maximum y-value of 1, and a minimum y-value of -1.

You should know what one cycle (period) of the graphs of the big three trig. functions looks like. They can be found by plotting points or graphing on a calculator. The following example figures depict the graphs of these functions: Sine, cosine, and tangent.

These graphs, like any other graphs of a function, can be transformed. The table in the accompanying figure outlines each change for each trig. ratio.

Pythagorean and Quotient Identities

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.

sin^2 x + cos^2 x = 1

1 + cot^2 x = csc^2 x

1 + tan^2 x = sec^2 x

Remember that sin^2 x = (sin x)^2.

Algebraic Manipulation of the Trigonometric Ratios

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.  Problems: Simplify (sin^2 x)(cos^2 x) + cos^4 x
   Solution: Factor.

             (cos^2 x)(sin^2 x + cos^2 x)

             Using a Pythagorean Identity, replace
             sin^2 x + cos^2 x with 1.

             (cos^2 x)(1)
             cos^2 x

Take the quiz on trigonometry. The quiz is very useful for either review or to see if you've really got the topic down.

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