A study guide that details the various aspects of trigonometry. From sine to tangent, this guide describes it all and explains it in simple terms.Full...
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Trigonometry Made Easy by Kevin Skolnick
Table of Contents Sine
3
Cosine
4
Tangent
5
Inverse vs. Reciprocal Functions
6
Radians and the Unit Circle
8
Graphs
9
Sine The sine of an angle is the ratio of the leg opposite that angle to the hypotenuse.
H y p o t e n u s e
A
Opposite
Example The triangle at right was set up using the pythagorean theorem. A = 30°
B = 60°
If the sine of A equals the opposite length over the hypotenuse, then the sine of 30° would equal 1/2.
A 30°
2
60°
1
B
Cosine The cosine of an angle is the ratio of the leg adjacent to that angle to the hypotenuse.
A Adjacent
H y p o t e n u s e
Example
A = 30°
B = 60°
If the cosine of A equals the opposite length over the hypotenuse, then the cosine of 30° would equal the square root of 3 over 2.
A 30°
2
60° Notice that cos(A) equals sin(B) and cos(B) equals sin(A). sin(A). The relationship between sine and cosine is as follows:
1
B
Tangent The tangent of an angle is the ratio of the leg opposite that angle to the adjacent leg.
A Adjacent
Opposite
Example
A = 30°
B = 60°
If the tangent of A equals the opposite length over the adjacent length, then the tangent of 30° would equal 1 over the square root of 3.
A 30°
2
60° As a side note:
1
B
Inverse vs. Reciprocal Functions They are not the same...
Inverse Functions If you take the sine of an angle, you get a ratio. However, you take the inverse sine (arcsine) of the ratio to get an angle . Inverse sine Inverse cosine Inverse tangent
= = =
arcsine arccosine arctangent
Notice the notations for the arcsine. This applies to all inverse trigonometric functions. Example:
Reciprocal Functions To reiterate, if you take the sine of an angle, you get opposite opposite over hypotenuse. If you take the reciprocal function (cosecant), you get hypotenuse over opposite . Reciprocal of sine Reciprocal of cosine Reciprocal of tangent
= = =
cosecant (csc) secant (sec) cotangent (cot)
Examples:
Inverse vs. Reciprocal Functions continued onto the next page
The Differences The inverse of a trigonometric function gives you the angle instead of the ratio. The reciprocal of a trigonometric function gives you the inverse of the ratio. This is better explained in the examples below:
Therefore,
Remember this:
IS NOT
This is why it is less confusing to use “arcsin” instead of “sin -1.”
Radians and the Unit Circle Radians are simply another unit to measure angles, like like degrees. Though, they are usually measured in terms of pi. Radians are used almost always instead of degrees in trigonometry.
The unit circle is a circle with a radius of 1. It demonstrates many ideas in trigonometry and in geometry. geometry. First, it can show the relationship between degrees and radians. The length of the segment between the two rays of any angle is equal to that angle s measure in radians. ʼ
45°
Also, the unit circle can define the functions of sine sine and cosine. If a triangle is drawn within the unit circle like the one below, its hypotenuse will always be 1 (the radius of the unit circle). Since the sine of an angle is the opposite leg over the hypotenuse and the cosine is the adjacent leg over the hypotenuse, the legs of the triangle below will be equal to the sine and cosine of the angle.