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Calculus ATrigonometric Functions

Dr. Bisher M. [email protected]

Department of MathematicsThe Islamic University of Gaza

2019-2020, Semester 1

Dr. Bisher M. Iqelan (IUG) Sec1.3: Trigonometric Functions 1st Semester, 2019-2020 1 / 14

Trigonometric Functions

The number of radians in the centralangle A′CB ′ within a circle of radiusr is de�ned as the number of"radius units" contained in the arc ssubtended by the central angle.With the central angle measuring θradians, this meansθ = s/r or s = rθ.

De�nition

♣

One complete revolution of the unit circle is 360◦ or 2π radians.Therefore π radians = 180◦ and

1 radian =180

π≈ 57.3◦ or 1◦ =

π

180≈ 0.017 radians.

Note



An angle in the xy−plane is in standard position if its vertex lies at theorigin and its initial ray lies along the positive x−axis. Angles measured coun-terclockwise from the positive x−axis are assigned positive measures; anglesmeasured clockwise are assigned negative measures.

De�nition

A central angle in a circle of radius 8 is subtended by an arc of length10π. Find the angle's radian in degree measures.Solution:

θ = 225◦

Example



Convert the following radiansto degrees:

1.π

4. 2.

π

6. 3.

π

3. 4.

4π

3

Solution:

1π4= �π

4.180�π

= 45◦

2π6= �π

6.180�π

= 30◦

3π3= �π

3.180�π

= 60◦

44π3

= 4�π3.180�π

= 240◦

Example 1

Convert the following degreesto radians:

1. 40◦. 2. 120◦. 3. 15◦. 4. 270◦

Solution:

1 40◦ = 40. π180

= 2π9

2 120◦ = 120. π180

= 2π3

3 15◦ = 15. π180

= π12

4 270◦ = 270. π180

= 3π2

Example 2



Let 0 < θ < π2

using the right triangle we de�ne the sixtrigonometric functions as follows

De�nition: Right Triangle De�nition



We de�ne the six trigonometric functions for any angle θ by �rstplacing the angle in standard position in a circle of radius r . Thende�ne the trigonometric functions in terms of the coordinates of thepoint P(x , y) where the angle's terminal ray intersects the circle asfollows:

sin θ = yr cos θ = x

r

sec θ = rx csc θ = r

y

tan θ = yx cot θ = x

y

De�nition: Circular De�nition

The coordinates of the point P(x , y) on the circle x2 + y2 = r2 canbe expressed in term of θ and r as x = r cos θ and y = r sin θ. Now,if r = 1 we see that x = cos θ and y = sin θ.

Note 1



By de�nition above, we immediately have the trig identities:

tan θ = sin θcos θ cot θ = 1

tan θ

sec θ = 1

cos θ csc θ = 1

sin θ

Note 2

Based on the special 30◦ − 60◦ − 90◦ and 45◦ − 45◦ − 90◦ righttriangles, we can deduce the following trig functions for the "specialangles" 30◦ = π

6, 45◦ = π

4, and 60◦ = π

3:

cos π6=√3

2cos π

4=√2

2cos π

3= 1

2

sin π6= 1

2sin π

4=√2

2sin π

3=√3

2

tan π6= 1√

3tan π

4= 1 tan π

3=√3

Note 3



In Figure 1 we see in whichquadrants or on which axes theterminal side of an angle0◦ ≤ theta < 360◦ may fall.

Figure: 1

Figure 2 summarizes the signs (positiveor negative) for the trigonometricfunctions based on the angle's quadrant:

Figure: 2

Note 4



Find the exact values of all six trigonometric functions of 120◦.

Solution:

We know 120◦ = 180◦ − 60◦. We can use thepoint (−1,

√3) on the terminal side of the

angle 120◦ in QII , since a basic right trianglewith a 60◦ angle has adjacent side of length 1,opposite side of length

√3, and hypotenuse of

length 2, as in the �gure below. Drawing thattriangle in QII so that the hypotenuse is on theterminal side of 120◦ makes r = 2, x = −1 andy =√3. Hence:

sin 120◦ = yr =

√3

2cos 120◦ = x

r = −12

tan 120◦ = yx =

√3

−1

csc 120◦ = ry = 2√

3sec 120◦ = r

x = 2

−1 cot 120◦ = xy = −1√

3

Example 1



Suppose that cos θ = − 4

5. Find the exact values of sin θ and tan θ.

Solution: Since cos θ = − 4

5, we can use 4 as the length of the adjacent side and 5 as

the length of the hypotenuse. The length of the opposite side must then be 3. Sincecos θ is negative, so θ must be in either QII or QIII as shown in Figure 3 below.

When θ is in QII , we see from Figure 3(a) that the point (−4, 3) is on the terminal

side of θ, and so we have x = −4, y = 3, and r = 5. Thus, sin θ = yr = 3

5and

tan θ = yx = 3

−4 .When θ is in QIII , we see from Figure 3(b) that the point (−4,−3) is on the terminal

side of θ, and so we have x = −4, y = −3, and r = 5. Thus, sin θ = yr = −3

5and

tan θ = yx = 3

4. Thus, either sin θ = 3

5and tan θ = − 3

4or sin θ = − 3

5and tan θ = 3

4.

Example 2


Periodicity and Graphs of the Trigonometric Functions

A function f (x) is periodic if there is a positive number p such thatf (x + p) = f (x) for every value of x . The smallest value of p is theperiod of f .

De�nition

The periods of sine, cosine, secant, and cosecant are each 2π.The periods of tangent and cotangent are both π. This leads tothe trigonometric identities:

sin(x + 2π) = sin x sin(x + 2π) = sin x

sec(x + 2π) = sec x csc(x + 2π) = csc x

tan(x + π) = tan x cot(x + π) = cot x

Note


Periodicity and Graphs of the Trigonometric Functions

The graphs of the six trigonometric functions are as follows(the shading indicates a single period):

Note



Trigonometric Identities



Example 2


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