College Trigonometry

College Trigonometry

George Voutsadakis1

1Mathematics and Computer Science Lake Superior State University

LSSU Math 131

George Voutsadakis (LSSU)

Trigonometry

January 2015 1 / 62

Outline

1 Trigonometric Identities and Equations Verification of Trigonometric Identities Sum, Difference and Cofunction Identities Double- and Half-Angle Identities Identities Involving Sum of Trigonometric Functions Inverse Trigonometric Functions Trigonometric Equations

George Voutsadakis (LSSU)

Trigonometry

January 2015 2 / 62

Trigonometric Identities and Equations Verification of Trigonometric Identities

Subsection 1 Verification of Trigonometric Identities

George Voutsadakis (LSSU)

Trigonometry

January 2015 3 / 62

Trigonometric Identities and Equations Verification of Trigonometric Identities

Review of Fundamental Trigonometric Identities

Reciprocal Identities:

sin x

=

1 csc

x

cos x

=

1 sec x

tan x

=

1 cot x

Ratio Identities:

tan

x

=

sin x cos x

cot x

=

cos x sin x

Pythagorean Identities: sin2 x + cos2 x = 1 tan2 x + 1 = sec2 x 1 + cot2 x = csc2 x

Odd-Even Identities:

sin (-x) = - sin x tan (-x) = - tan x sec (-x) = sec x cos (-x) = cos x cot (-x) = - cot x csc (-x) = - csc x

George Voutsadakis (LSSU)

Trigonometry

January 2015 4 / 62

Trigonometric Identities and Equations Verification of Trigonometric Identities

General Guidelines for Verifying Trigonometric Identities

If one side is more complex, try to simplify it to match the simpler side;

Perform indicated operations (e.g., adding fractions or expanding powers) and be aware of potential factorizations;

Use previously established identities that allow rewriting expressions in equivalent forms;

Rewrite one side so that it involves only sines and cosines;

Rewrite one side in terms of single trigonometric function;

Multiplying both numerator and denominator of a fraction by the same expression may be useful;

Keeping the final goal in mind is paramount: Does it involve products, quotients, sums, radicals, powers? Does the form provide insight on the most likely way to reach the goal?

Proving an identity is, sometimes, partly art and partly science!

George Voutsadakis (LSSU)

Trigonometry

January 2015 5 / 62

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