Calculus with business applications, Lehigh U, Lecture 0 notes Summer 0 Trigonometric functions. Trigonometric functions often arise in physical applications with periodic motion. They do not arise often in business eamples. We will still cover them but with less detail than other functions.. Use radian measure rather than degrees for calculus since this makes computation of derivatives easier. See the unit circle diagram tet page 3. For radians we specify an angle in terms terms of the arc length along the unit circle, a circle of radius centered at the origin. For eample, the angle formed by the positive -ais and the positive y-ais (horizontal line going right and vertical line going up) is 90 degrees. The corresponding arc on the circle is /4 th of the circumference, π. So the radian measure of a 90 degree angle is 4 π = π. In a similar manner the ratio of d degrees to a complete turn of 360 degrees is the same as the ratio of the radian measure of the angle to going around the unit circle once, π. So we have the conversion d in degrees is s = π = π d in radians. 360 80 3. Values of cos and sin are the and y coordinates respectively of point on the unit circle at distance along the circle from (, 0). See the figure tet page 36 for basic values you should know. Note that you really only need to memorize 4 values: 0 degrees: (cos 0, sin 0) = (, 0) 30 degree angles: (cos π, sin π) = ( 3, ) 6 6 4 degrees: (cos π, sin π) = (, ) 4 4 60 degrees: (cos π, sin π) = (, 3 ) 3 3 The rest can be obtained by symmetry 4. The notation etc represents a function. That is, sin π is the value of the sine 6 function evaluated at the angle π. It is not sin times π. In particular we cannot 6 6 distribute, sin( + y) is not + sin y. Also sin by itself has no meaning. It is a function and needs and argument. If you find yourself writing just sin you are doing something wrong.
Calculus with business applications, Lehigh U, Lecture 0 notes Summer 0 We use particular shorthand notation for powers, writing, for eample, sin for (). We can do this for any power ecept, sin represents the inverse function to sine (described below). There is a special term for the reciprocal of the sine function () = = csc, the cosecant function. Similarly, secant is the reciprocal of cosine, = sec and cotangent is the reciprocal of tangent, = cot. cos tan Recall also that tangent is the ratio of sine to cosine: tan = so cotangent is the cos ratio of cosine to sine, cot = cos.. Using the Pythagorean theorem on a triangle with hypotenuse yields the basic identity sin + cos =. As other trigonometric functions are defined in terms of sin and cos we can obtain other Pythagorean identities. Tet page 37. For eample, to show the identity + cot = csc multiply sin + cos = by cos and use cot = and csc =. sin sin sin ( sin sin + cos = ) + cos = + sin sin cot = csc Be aware that double and half angle identities eist but do not memorize them. We will not use them in Math 8. If you ever need them you can look them up. 6. If cos() = 4/ for 0 < < π/, use basic trigonometric identities and the unit circle to determine sin(), tan(), cos(π/ ), cos(π ), cos(π + ) and sin( ). Since = sin () + cos () = sin () + (4/) we get sin () = (6/) = 9/. Hence sin() = ±3/ and since 0 < < π/, sin() > 0 so we have sin() = 3/. tan() = sin() = 3/ = 3/4. Using the similar triangles shown below for the various cos() 4/ angles we get cos(π/ ) = 3/, cos(π ) = 4/, cos(π + ) = 4/ and sin( ) = 3/. (0,) ( 4, 3 ) π + (0,) ( 3, 4 ) ( 4, 3 ) (0,) π ( 4, 3 ) π ( 4, 3 ) 7. Identities that follow from moving a basic triangle on the unit circle in a manner similar to the previous eample include: (a) sin(π ) = sin() and cos(π ) = cos() (b) sin( ) = sin() and cos( ) = cos() (c) cos() = sin( + π )
Calculus with business applications, Lehigh U, Lecture 0 notes Summer 0 3 8. Inverse trigonometric functions are described in the tet pages 39-40. What angles have sin =? One answer is π/4. But also..., 4π + π/4, π + π/4, π/4, π + π/4, 4π+π/4,... as well as..., 4π+3π/4, π+3π/4, 3π/4, π+3π/4, 4π+3π/4,.... To define the inverse function we pick a particular value. For inverse sine functions we use angles between π/ and π/ and for inverse cosine functions we use angles between 0 and π. Recall the notation for inverse functions from lecture 0. Inverse trigonometric functions are denoted, for eample, sin (the term arcsine is also used). Note that this is not. It is the angle between π/ and π/ such that sin =. 9. Derivatives of inverse trigonometric functions arise in engineering applications. To determine these we will use computations like the following eample and eample 4b in the tet page 4. Simplify tan(cos ) Let = cos, then cos =. This is adjacent/hypotenuse in the triangle below. Using the Pythagorean Theorem gives the remaining side as =. Then tan(cos ) = tan = as it is opposite/adjacent in the triangle. =
Calculus with business applications, Lehigh U, Lecture 0 notes Summer 0 4 Supplementary problems Degrees and radians P. Convert the following radian measures to degrees: π 3, 7π 6, π 4, π P. Convert the following degrees to radians: 0, 7, 0, 80 Basic trigonometric values P.3 Find sin 4π, cos 4π, tan 4π 3 3 3 and sin 7π 6, cos 7π 6, tan 7π 6 Trigonometric identities P.4 Derive the trigonometric identity tan + = sec from sin + cos = and the definitions of tan and sec in terms of sin and cos P. If sin = 3 and 0 π determine cos, tan, cot, sec, csc P.6 If sin = and 0 π determine cos( + π) and cos( π ). Eplain your 4 computations using a unit circle. P.7 Use a right triangle to write the epression in terms of : sin ( ) cos 3
Calculus with business applications, Lehigh U, Lecture 0 notes Summer 0 Solutions to supplementary problems Degrees and radians S. We get the following degrees π 80 = 0, 7π 80 = 0, π 80 3 π 6 π 4 π = 4, π 80 π = S. We get the following radians 0 π 80 = π 8, 7 π 80 = π, 0 π 80 = π 6, 80 π 80 = π Basic trigonometric values S.3 See the unit circle diagram in the tet: sin 4π = 3, cos 4π =, tan 4π = 3 and 3 3 3 sin 7π =, cos 7π = 3, tan 7π = 6 6 6 3 Trigonometric identities S.4 To show the identity tan + = sec multiply sin + cos = by tan = sin and sec =. cos cos ( cos sin + cos = ) sin + = cos cos tan + = sec S. Use the Pythagorean theorem sin + cos = with sin = ( : 3 ) 3 + cos = cos = = 44 cos = 44/69 = 69 69 We take the positive root since cosine is positive for 0 < < π. Then, from the definitions: tan = sin = /3 = cos /3, cot = cos sin = /3 /3 =, sec = cos = /3 = 3, csc = sin = /3 = 3. 3. cos and use S.6 Use the Pythagorean theorem sin + cos = with sin = : ( 4 4) + cos = cos = = cos = /6 =. 6 6 4 We take the positive root since cosine is positive for 0 < < π. Similar triangles in the figure show cos( + π) = 4 and cos( π ) = 4. S.7 Let = cos 3 then cos = 3. This is adjacent/hypotenuse in the triangle below. Using the Pythagorean Theorem gives the remaining side as 3 = 9. Then sin ( cos ) 3 = sin = 9 as it is opposite/hypotenuse in the triangle.