Cheatography
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A quick reference guide for solving trigonometric proofs and some basic trig identities to use.
Solving Trigonometric Proofs
Method 1 |
Pick one side of the equation (usually the most complicated side) and work with that side until it is equal to the other side |
Method 2 |
Work with both sides simultaneously until they are both equal to the same expression. |
Helpful Techniques
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Rewrite Rewrite the expression in terms of sine and cosine only
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Multiply by One Multiply the numerator and denominator of a rational expression by a carefully chosen "one"
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Combine fractions Write sums and differences of rational expressions as a single fraction
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Factor Factor trigonometric expressions, using "u-substitution" as needed
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Pythagorean Theorem Look for combinations or portions of Pythagorean Identities. Remember that you can multiply, divide, add or subtract the identity to get a new version.
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Goal Always keep the goal in mind. As you manipulate one side of the equation, keep the other side in mind. Look for commonalities
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Verifying Trigonometric Identities
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Pythagorean Identities
sin2x + cos2x = 1
sin2x = 1 - cos2x
cos2x = 1 - sin2x
1 + cot2x = csc2x
cot2x = csc2x - 1
tan2x + 1 = sec2x
tan2x = sec2x - 1 |
Divide original Pythagorean identity by sin2x or cos2x to get other identities, subtract to get even more.
Basic Trigonometric Functions
sin (-x) = -sin (x) |
tan (x) = sin (x) / cos (x) |
csc (x) = 1/sin (x) |
cos (-x) = cos (x) |
cot (x) = cos (x) / sin (x) |
sec (x) = 1/cos (x) |
tan (-x) = -tan (x) |
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cot (x) = 1/tan (x) |
Complementary Angle Identities
sin (pi/2 - x) = cos (x) |
cos (pi/2 - x) = sin (x) |
tan (pi/2 - x) = cot (x) |
Sum and Difference Identities
sin (a +/- b) = |
sin (a) cos (b) +/- cos (a) sin (b) |
cos (a +/- b) = |
cos (a) cos (b) -/+ sin (a) sin (b) |
Use sin (a +/- b) / cos (a +/- b) to find tan (a +/- b)
Double Angle Identities
sin (2x) |
cos (2x) |
2 sin (x) cos (x) |
cos2 (x) - sin2 (x) |
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1 - 2 sin2 (x) |
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2 cos2 (x) - 1 |
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