Definition of the Trigonometric Functions: Cheat Sheet
Trig Cheat Sheet
Definition of the Trig Functions
Right triangle definition For this definition we assume that p 0 < q < or 0° < q < 90° . 2 Unit circle definition For this definition q is any angle. y ( x, y ) hypotenuse opposite y 1 x
q x q adjacent sin q = opposite hypotenuse adjacent cos q = hypotenuse opposite tan q = adjacent hypotenuse opposite hypotenuse sec q = adjacent adjacent cot q = opposite csc q = sin q = y =y 1 x cos q = = x 1 y tan q = x 1 y 1 sec q = x x cot q = y csc q =
Facts and Properties
Domain The domain is all the values of q that can be plugged into the function. sin q , q can be any angle cos q , q can be any angle 1ö æ tan q , q ¹ ç n + ÷ p , n = 0, ± 1, ± 2,K 2ø è csc q , q ¹ n p , n = 0, ± 1, ± 2,K 1ö æ sec q , q ¹ ç n + ÷ p , n = 0, ± 1, ± 2,K 2ø è cot q , q ¹ n p , n = 0, ± 1, ± 2,K
Period The period of a function is the number, T, such that f (q + T ) = f (q ) . So, if w is a fixed number and q is any angle we have the following periods. sin ( wq ) ® cos (wq ) ® tan (wq ) ® csc (wq ) ® sec (wq ) ® cot (wq ) ® T= T T T T T 2p w 2p = w p = w 2p = w 2p = w p = w
Range The range is all possible values to get out of the function. csc q ³ 1 and csc q £ -1 -1 £ sin q £ 1 -1 £ cos q £ 1 sec q ³ 1 and sec q £ -1 -¥ < tan q < ¥ -¥ < cot q < ¥
© 2005 Paul Dawkins
Formulas and Identities
Tangent and Cotangent Identities sin q cos q tan q = cot q = cos q sin q Reciprocal Identities 1 1 csc q = sin q = sin q csc q 1 1 sec q = cos q = cos q sec q 1 1 cot q = tan q = tan q cot q Pythagorean Identities sin 2 q + cos 2 q = 1 tan 2 q + 1 = sec 2 q 1 + cot 2 q = csc 2 q Even/Odd Formulas sin ( -q ) = - sin q csc ( -q ) = - csc q cos ( -q ) = cos q tan ( -q ) = - tan q Periodic Formulas If n is an integer. sin (q + 2p n ) = sin q tan (q + p n ) = tan q sin ( 2q ) = 2sin q cos q cos ( 2q ) = cos 2 q - sin 2 q = 2 cos 2 q - 1 = 1 - 2sin 2 q 2 tan q tan ( 2q ) = 1 - tan 2 q Degrees to Radians Formulas If x is an angle in
Definition of the Trig Functions
Right triangle definition For this definition we assume that p 0 < q < or 0° < q < 90° . 2 Unit circle definition For this definition q is any angle. y ( x, y ) hypotenuse opposite y 1 x
q x q adjacent sin q = opposite hypotenuse adjacent cos q = hypotenuse opposite tan q = adjacent hypotenuse opposite hypotenuse sec q = adjacent adjacent cot q = opposite csc q = sin q = y =y 1 x cos q = = x 1 y tan q = x 1 y 1 sec q = x x cot q = y csc q =
Facts and Properties
Domain The domain is all the values of q that can be plugged into the function. sin q , q can be any angle cos q , q can be any angle 1ö æ tan q , q ¹ ç n + ÷ p , n = 0, ± 1, ± 2,K 2ø è csc q , q ¹ n p , n = 0, ± 1, ± 2,K 1ö æ sec q , q ¹ ç n + ÷ p , n = 0, ± 1, ± 2,K 2ø è cot q , q ¹ n p , n = 0, ± 1, ± 2,K
Period The period of a function is the number, T, such that f (q + T ) = f (q ) . So, if w is a fixed number and q is any angle we have the following periods. sin ( wq ) ® cos (wq ) ® tan (wq ) ® csc (wq ) ® sec (wq ) ® cot (wq ) ® T= T T T T T 2p w 2p = w p = w 2p = w 2p = w p = w
Range The range is all possible values to get out of the function. csc q ³ 1 and csc q £ -1 -1 £ sin q £ 1 -1 £ cos q £ 1 sec q ³ 1 and sec q £ -1 -¥ < tan q < ¥ -¥ < cot q < ¥
© 2005 Paul Dawkins
Formulas and Identities
Tangent and Cotangent Identities sin q cos q tan q = cot q = cos q sin q Reciprocal Identities 1 1 csc q = sin q = sin q csc q 1 1 sec q = cos q = cos q sec q 1 1 cot q = tan q = tan q cot q Pythagorean Identities sin 2 q + cos 2 q = 1 tan 2 q + 1 = sec 2 q 1 + cot 2 q = csc 2 q Even/Odd Formulas sin ( -q ) = - sin q csc ( -q ) = - csc q cos ( -q ) = cos q tan ( -q ) = - tan q Periodic Formulas If n is an integer. sin (q + 2p n ) = sin q tan (q + p n ) = tan q sin ( 2q ) = 2sin q cos q cos ( 2q ) = cos 2 q - sin 2 q = 2 cos 2 q - 1 = 1 - 2sin 2 q 2 tan q tan ( 2q ) = 1 - tan 2 q Degrees to Radians Formulas If x is an angle in