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0.1 Basic functions and identities

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Graphs of the parent functions

Three graphs side-by-side. From left to right, graph of the identify function, square function, and square root function. All three graphs extend from -4 to 4 on each axis.
Three graphs side-by-side. From left to right, graph of the cubic function, cube root function, and reciprocal function. All three graphs extend from -4 to 4 on each axis.
Three graphs side-by-side. From left to right, graph of the absolute value function, exponential function, and natural logarithm function. All three graphs extend from -4 to 4 on each axis.

Graphs of the trigonometric functions

Three graphs of trigonometric functions side-by-side. From left to right, graph of the sine function, cosine function, and tangent function. Graphs of the sine and cosine functions extend from negative two pi to two pi on the x-axis and two to negative two on the y-axis. Graph of tangent extends from negative pi to pi on the x-axis and four to negative 4 on the y-axis.
Three graphs of trigonometric functions side-by-side. From left to right, graph of the cosecant function, secant function, and cotangent function. Graphs of the cosecant function and secant function extend from negative two pi to two pi on the x-axis and ten to negative ten on the y-axis. Graph of cotangent extends from negative two pi to two pi on the x-axis and twenty-five to negative twenty-five on the y-axis.
Three graphs of trigonometric functions side-by-side. From left to right, graph of the inverse sine function, inverse cosine function, and inverse tangent function. Graphs of the inverse sine and inverse tangent extend from negative pi over two to pi over two on the x-axis and pi over two to negative pi over two on the y-axis. Graph of inverse cosine extends from negative pi over two to pi on the x-axis and pi to negative pi over two on the y-axis.
Three graphs of trigonometric functions side-by-side. From left to right, graph of the inverse cosecant function, inverse secant function, and inverse cotangent function.

Trigonometric identities

Pythagorean Identities cos 2 t + sin 2 t = 1 1 + tan 2 t = sec 2 t 1 + cot 2 t = csc 2 t
Even-Odd Identities cos (− t ) = cos   t sec (− t ) = sec   t sin (− t ) = − sin   t tan (− t ) = − tan   t csc (− t ) = − csc   t cot (− t ) = − cot   t
Cofunction Identities cos   t = sin ( π 2 − t ) sin   t = cos ( π 2 − t ) tan   t = cot ( π 2 − t ) cot   t = tan ( π 2 − t ) sec   t = csc ( π 2 − t ) csc   t = sec ( π 2 − t )
Fundamental Identities tan   t = sin   t cos   t sec   t = 1 cos   t csc   t = 1 sin   t cot   t = 1 tan   t = cos   t sin   t
Sum and Difference Identities cos ( α + β ) = cos   α   cos   β − sin   α   sin   β cos ( α − β ) = cos   α   cos   β + sin   α   sin   β sin ( α + β ) = sin   α   cos   β + cos   α   sin   β sin ( α − β ) = sin   α   cos   β − cos   α   sin   β tan ( α + β ) = tan   α + tan   β 1 − tan   α   tan   β tan ( α − β ) = tan   α − tan   β 1 + tan   α   tan   β
Double-Angle Formulas sin ( 2 θ ) = 2   sin   θ   cos   θ cos ( 2 θ ) = cos 2 θ − sin 2 θ cos ( 2 θ ) = 1 − 2   sin 2 θ cos ( 2 θ ) = 2   cos 2 θ − 1 tan ( 2 θ ) = 2   tan   θ 1 − tan 2 θ
Half-Angle Formulas sin   α 2 = ± 1 − cos   α 2 cos   α 2 = ± 1 + cos   α 2 tan   α 2 = ± 1 − cos   α 1 + cos   α tan   α 2 = sin   α 1 + cos   α tan   α 2 = 1 − cos   α sin   α
Reduction Formulas sin 2 θ = 1 − cos ( 2 θ ) 2 cos 2 θ = 1 + cos ( 2 θ ) 2 tan 2 θ = 1 − cos ( 2 θ ) 1 + cos ( 2 θ )
Product-to-Sum Formulas cos   α   cos   β = 1 2 [ cos ( α − β ) + cos ( α + β ) ] sin   α   cos   β = 1 2 [ sin ( α + β ) + sin ( α − β ) ] sin   α   sin   β = 1 2 [ cos ( α − β ) − cos ( α + β ) ] cos   α   sin   β = 1 2 [ sin ( α + β ) − sin ( α − β ) ]
Sum-to-Product Formulas sin   α + sin   β = 2   sin ( α + β 2 )   cos ( α − β 2 ) sin   α − sin   β = 2   sin ( α − β 2 )   cos ( α + β 2 ) cos   α − cos   β = − 2   sin ( α + β 2 )   sin ( α − β 2 ) cos   α + cos   β = 2   cos ( α + β 2 )   cos ( α − β 2 )
Law of Sines sin   α a = sin   β b = sin   γ c a sin   α = b sin   β = c sin   γ
Law of Cosines a 2 = b 2 + c 2 − 2 b c   cos   α b 2 = a 2 + c 2 − 2 a c   cos   β c 2 = a 2 + b 2 − 2 a b   cos   γ

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
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John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
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emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
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Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
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Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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