Trigonometric Functions By Daria Eiteneer Topics Covere: Reminer: relationship between egrees an raians The unit circle Definitions of trigonometric functions for a right triangle Definitions of trigonometric functions for a unit circle Exact values for trigonometric functions of most commonly use angles Trigonometric functions of any angle θ' in terms of angle θ in quarant I Trigonometric functions of negative angles Some useful relationships among trigonometric functions Double angle formulas Half angle formulas Angle aition formulas Sum, ifference an prouct of trigonometric functions Graphs of trigonometric functions Inverse trigonometric functions Principal values for inverse trigonometric functions Relations between inverse trigonometric functions Graphs of inverse trigonometric functions Using trigonometric functions: components of a vector Using trigonometric functions: phase shift of a wave Derivatives of trigonometric functions Note: All figures, unless otherwise specifie, have a permission to be copie, istribute an/or moifie uner the terms of the GNU Free Documentation License, Version 1. or later.
Reminer: Relationship Between Degrees an Raians A raian is efine as an angle θ subtene at the center of a circle for which the arc length is equal to the raius of that circle (see Fig.1). Fig.1. Definition of a raian. The circumference of the circle is equal to πr, where R is the raius of the circle. Consequently, 360 = π raians. Thus, 1 raian=360 / π 57.96 1 = ( π /360) raians 0.01745 raians The Unit Circle In mathematics, a unit circle is efine as a circle with a raius of 1. Often, especially in applications to trigonometry, the unit circle is centere at the origin (0,0) in the coorinate plane. The equation of the unit circle in the coorinate plane is x + y = 1. As mentione above, the unit circle is taken to be 360, or π raians. We can ivie the coorinate plane, an therefore, the unit circle, into 4 quarants. The first quarant is efine in terms of coorinates by x>0, y>0, or, in terms of angles, by 0 < θ<90, or 0< θ< π/. The secon quarant is efine by x<0, y>0, or 90 < θ<180, or π/< θ< π. The thir quarant is efine by x<0, y<0, or 180 < θ<70, or π< θ<3π/. Finally, the fourth quarant is efine by x>0, y<0, or 70 < θ<360, or 3 π/< θ< π.
Trigonometric Functions Definitions of Trigonometric Functions For a Right Triangle A right triangle is a triangle with a right angle (90 ) (See Fig.). Fig.. Right triangle. For every angle θ in the triangle, there is the sie of the triangle ajacent to it (from here on enote as aj ), the sie opposite of it (from here on enote as opp ), an the hypotenuse (from here on enote as hyp ), which is the longest sie of the triangle locate opposite of the right angle. For angle θ, the trigonometric functions are efine as follows: opp sine of θ = sin θ = hyp aj cosine of θ = cos θ = hyp tangent of θ = tan θ = cotangent of θ = cot θ = secant of θ = sec θ = cosecant of θ = csc θ = sinθ cosθ = opp aj 1 tanθ = cosθ sinθ = aj opp 1 cosθ = hyp aj 1 sinθ = hyp opp
Definitions of Trigonometric Functions For a Unit Circle In the unit circle, one can efine the trigonometric functions cosine an sine as follows. If (x,y) is a point on the unit cirlce, an if the ray from the origin (0,0) to that point (x,y) makes an angle θ with the positive x-axis, (such that the counterclockwise irection is consiere positive), then, cos θ = x/1 = x sin θ = y/1 = y Then, each point (x,y) on the unit circle can be written as (cos θ, sin θ). Combine with the equation x + y = 1, the efinitions above give the relationship sin θ+cos θ=1. In aition, other trigonometric functions can be efine in terms of x an y: tan θ = sin θ/cos θ = y/x cot θ = cos θ/sin θ = x/y sec θ = 1/cos θ = 1/x csc θ = 1/ sin θ = 1/y Fig.3 below shows a unit circle in the coorinate plane, together with some useful values of angle θ, an the points (x,y)=(cos θ, sin θ), that are most commonly use (also see table in the following section). Fig.3. Most commonly use angles an points of the unit circle. Note: For θ in quarant I, sinθ>0, cos θ >0; for θ in quarant II, sin θ>0, cos θ<0; for θ in quarant III, sin θ<0, cos θ<0; an for θ in quarant IV, sin θ
Exact Values for Trigonometric Functions of Most Commonly Use Angles θ in egrees θ in raians sinθ cosθ tanθ 0 0 0 1 0 30 π 6 1 3 3 3 45 π 4 1 60 π 3 1 3 3 90 π 1 0 unefine 180 π 0-1 0 70 3 π -1 0 unefine 360 π 0 1 0 Note: Exact values for other trigonometric functions (such as cotθ, secθ, an cscθ) as well as trigonometric functions of many other angles can be erive by using the following sections. Trigonometric Functions of Any Angle θ' in Terms of Angle θ in Quarant I θ' sinθ' cosθ' tanθ' θ' sinθ' cosθ' tanθ' 90 +θ π/+θ cosθ -sinθ cotθ 90 -θ π/ θ cosθ sinθ cotθ 180 +θ π+θ sinθ cosθ tanθ 180 -θ π-θ sinθ cosθ tanθ 70 +θ 3π/+θ cosθ sinθ cotθ 70 -θ 3π/ θ cosθ sinθ cotθ k(360 )+θ k(360 )-θ k(π)+θ sinθ cosθ tanθ k(π)-θ sinθ cosθ tanθ k=integer k=integer
Trigonometric Functions of Negative Angles sin(- θ) = -sinθ cos(- θ) = cosθ tan(- θ) = -tanθ Some Useful Relationships Among Trigonometric Functions sin θ + cos θ = 1 sec θ tan θ = 1 csc θ cot θ = 1 Double Angle Formulas sin θ = sin θ cosθ cos θ = cos θ sin θ = 1- sin θ = cos θ -1 tanθ tanθ = 1 tan θ Half Angle Formulas Note: in the formulas in this section, the + sign is use in the quarants where the respective trigonometric function is positive for angle θ/, an the - sign is use in the quarants where the respective trigonometric function is negative for angle θ/. sin θ = ± 1 cosθ cos θ = ± 1 cosθ tan θ = ± 1 cosθ 1 cosθ = sinθ 1 cosθ = 1 cosθ sinθ
Angle Aition Formulas Note: in this an the following section, letters A an B are use to enote the angles of interest, instea of the letter θ. sin A±B = sinacosb ± cosa sinb cos A± B = cosa cos B sina sinb tana± tanb tan A± B = 1 tana tanb cot(a±b)= cotacotb 1 cotb ± cota Sum, Difference an Prouct of Trigonometric Functions sina + sinb = sin A B sina sinb = sin A B cosa + cosb = cos A B cosa cosb = sin A B cos A B cos A B cos A B sin A B sina sinb = cosa cosb = sina cosb = 1 [cos A B cos A B ] 1 [cos A B cos A B ] 1 [sin A B sin A B ]
Graphs of Trigonometric Functions (Fig.4, a-f) Note: In each graph in Fig.4, the horizontal axis (x) is measure in raians. Ref. Weisstein, Eric W. Sine. Cosine. Tangent. Cotangent. Secant. Cosecant. From MathWorl A Wolfram Web Resource: http://mathworl.worlfram.com Fig.4a. Graph of sin(x). Fig.4b. Graph of cos(x). Fig.4c. Graph of tan(x). Fig.4. Graph of cot(x). Fig.4e. Graph of sec(x). Fig.4f. Graph of csc(x).
Inverse Trigonometric Functions Inverse Trigonometric Functions If x=sin(y), then y=sin -1 (x), i.e. s is the angle whose sine is y. In other wors, x is the inverse sine of y. Another name for inverse sine is arcsine, an the notation use is y=arcsin(x). Similarly, we can efine inverse cosine, inverse tangent, inverse cotangent, inverse secant an inverse cosecant. All of the inverse functions are many-value functions of x (for each value of x, there are many corresponing values of y), which are collections of single-value functions (for each value of x, there is only one corresponing value of y) calle branches. For many purposes a particular branch is require. This is calle the principal branch an the values for this branch are calle principal values. x = sin(y) y = sin 1 (x) = arcsin(x) x = cos(y) y = cos 1 (x) = arccos(x) x = tan(y) x = cot(y) y = tan 1 (x) = arctan(x) y = cot 1 (x) = arccot(x) x = sec(y) y = sec 1 (x) = arcsec(x) x = csc(y) y = csc 1 (x) = arccsc(x) Principal Values for Inverse Trigonometric Functions Principal Values for x 0 0 sin 1 x π/ 0 cos 1 x π/ 0 tan 1 x π/ 0 cot 1 x π/ 0 sec 1 x π/ 0 csc 1 x π/ Principal Values for x 0 π/ sin 1 x 0 π/ cos 1 x π π/ tan 1 x 0 π/ cot 1 x π π/ sec 1 x π π/ csc 1 x 0
Relations Between Inverse Trigonometric Functions Note: In all cases, it is assume that principal values are use. sin -1 x + cos -1 x = π/ tan -1 x + cot -1 x = π/ sec -1 x +csc -1 x = π/ csc -1 x = sin -1 (1/x) sec -1 x = cos -1 (1/x) cot -1 x = tan -1 (1/x) sin -1 (-x) = -sin -1 x cos -1 (-x) = π - cos-1 x tan -1 (-x) = -tan -1 x cot -1 (-x) = π cot-1 x sec -1 (-x) = π sec-1 x csc -1 (-x) = -csc -1 x Graphs of Inverse Trigonometric Functions (Fig.5, a-f) Note: In each graph in Fig.5, the vertical axis (y) is measure in raians. Only portions of curves corresponing to principal values are shown. Ref. Weisstein, Eric W. Inverse Sine. Inverse Cosine. Inverse Tangent. Inverse Cotangent. Inverse Secant. Inverse Cosecant. From MathWorl A Wolfram Web Resource: http://mathworl.worlfram.com Fig.5a. Graph of sin -1 (x). Fig.5b. Graph of cos -1 (x).
Fig.5c. Graph of tan -1 (x). Fig.5. Graph of cot -1 (x). Fig.5e. Graph of sec -1 (x). Fig.5f. Graph of csc -1 (x).
Using Trigonometric Functions Resolving Vectors into Components: The geometric way of aing vectors is not recommene whenever great accuracy is require or in three-imensional problems. In such cases, it is better to make use of the projections of vectors along coorinate axis, also known as components of the vector. Any vector can be completely escribe by its components. Consier an arbitrary vector A (from now on, the bol-case letters are use to signify vectors, whereas the regular-font letter A signifies the length of the vector A) lying in the xy-plane an making an arbitrary angle θ with the positive x-axis, as shown in Fig.6: Fig.6. Arbitrary vector A in the xy-plane. This vector A can be expresse as the sum of two other vectors, A x an A y. From Fig.6, it can be seen that the three vectors form a right triangle an that A=A x +A y. It is conventional to refer to the components of vector A, written A x an A y (without the bolface notation). The component A x represents the projection of A along the x-axis, an the component A y represents the projection of A along the y-axis. These components can be positive or negative. From Fig.6 an the efinition of sine an cosine for a right triangle, we see that cos θ=ax/a an sin θ=ay/a. Hence, the components of A are
A x =A cosθ A y =A sinθ These components from the two sies of a right triangle with hypotenuse of length A. Thus, it follows that the magnitue an irection of A are relate to its components through the expressions A= Ax Ay θ=tan 1 Ay Ax Note that the signs of the comonents A x an A y epen on the angle θ. For example, if θ =10, then A x is negative an A y is positive. When solving problems, one can specify a vector A either with its components Ax an Ay or with its magnitue an irection A an θ. Furthermore, one can express a vector A as A=Ax x Ay y=a cosθ x sinθ y, where x an y are unit vectors (length of one) in the irection of x- an y-axis, respectively. Writing a Phase Shift of a Wave Waves can have many ifferent shape. One of the simplest to eal with an also one that is of a particular interest when it comes to simple harmonic motion is a sinusoial wave. It is calle sinusoial because the shape that the wave takes either in space or in time is that of a sine curve (see Fig.7a,b): Fig.7.(a) A one-imensional sinusoial wave in space. Fig.7b. A one-imensional sinusoial wave in time.
A function escribing such a wave can be written as y=asin π λ x± π T t φ, where A is the amplitue of the wave, λ is the wavelength of the wave, T is the perio of the wave, an φ is the phase shift of the wave. The ± sign in front of the secon term in the parentheses epens on whether the wave is moving to the right (-) or to the left (+). The entire expression in the parentheses can also be written as Φ(x,t) an is calle the total phase of the wave, an is a function of position x an time t, in contrast to the phase shift φ, which is a constant. Then, the entire expression becomes y = A sin Φ(x,t). Because the sine function is perioic with perio π, sin( Φ+πn)=sin( Φ) for any integer n. φ is referre to as a phase-shift, because it represents a "shift" from zero phase. But a change in φ is also referre to as a phase-shift. Fig.8 shows two curves: the re one with zero phase, an the blue one with a non-zero phase. Fig.8. Phase shift. Two oscillators that have the same frequency an same wavelength, will have a phase ifference, if their phase shift φ is ifferent. If that is the case, the oscillators are sai to be out of phase with each other. The amount by which such oscillators are out of phase with each other can be expresse in egrees from 0 to 360, or in raians from 0 to π (see Fig.9a,b). Fig.9a. Waves that are in phase. Fig.9b. Waves that are out of phase.
Derivatives of Trigonometric an Inverse Trigonometric Functions sin x =cos x x cos x = sin x x x tan x =sec x x cot x = csc x sec x =sec x tan x x csc x = csc x cot x x x sin 1 x = 1 π, 1 x sin 1 x π x cos 1 x = 1, 0 cos 1 x π 1 x x tan 1 x = 1 π, 1 x tan 1 x π x cot 1 x = 1, 0 cot 1 x π 1 x x sec 1 x = ±1 x x 1, + if 0 sec 1 x π, - if π sec 1 x π x csc 1 x = 1 x x 1, - if 0 csc 1 x π, + if π csc 1 x 0