Mathematics

Lecture. 3

Chapter. 1

Trigonometric Functions

By

Dr. Mohammed Ramidh

Trigonometric Functions This section reviews the basic trigonometric functions. Trigonometric functions are important because they are periodic.

• Radian Measure:

The radian measure of the angle ACB at the center of the unit circle equals the

length of the arc that ACB cuts from the unit circle. Figure shows that s = r θ

is the length of arc cut from a circle of radius r when the subtending angle θ

π radians = 180°.

Degrees to radians: multiply by 𝜋

180

Radians to degrees: multiply by 180

𝜋

For example: 45° in radian measure is → 45◦ ∗𝜋

180 =

𝜋

4

𝜋

6 in degree measure is →

𝜋

6 *

180

𝜋 = 30 °.

”Figure shows the angles of two common triangles in both measures.

Angles measured counterclockwise from the positive x-axis are assigned positive measures;

”angles measured clockwise are assigned negative measures, shows in Figure.

• Types and trigonometric relations:

The Six Basic Trigonometric Functions,

sin θ = opphyp

cos θ = adjhyp

tan θ = opp adj

cot θ = adjopp

sec θ = hypadj

csc θ = hypopp

”the angle in standard position in a circle of radius r. We then define the trigonometric functions in terms of the coordinates of the point P(x, y) where the angle’s terminal ray intersects the circle (shows in Figure).

◦ sine: sin θ = 𝑦

𝑟 cosine: cos θ =

𝑥

𝑟

◦ tangent: tan θ = 𝑦

𝑥 cotangent: cot θ =

𝑥

𝑦

◦ secant: sec θ = 𝑟

𝑥 cosecant: csc θ =

𝑟

𝑦

Notice also the following definitions, whenever the quotients are defined.

tan θ = sin θ

cos θ cot θ =

1

tan θ

sec θ = 1

cos θ csc θ =

1

sin θ

”The exact values of these trigonometric ratios for some angles can be read from the triangles in Figure.

sin 𝛑

4 =

1

2 sin 𝛑

6 =

1

2 sin 𝛑

3 =

3

2

cos 𝛑

4=

1

2 cos 𝛑

6=

3

2 cos 𝛑

3 =

1

2

tan 𝛑

4 = 1 tan 𝛑

6 =

1

3 tan 𝛑

3 = 3

”The CAST rule (shown in Figure) is useful for remembering when the basic trigonometric functions are positive or negative.

” from the triangle in Figure, we see that.

sin 2𝛑

3 =

3

2 cos 2𝛑

3 = −

1

2

tan 2𝛑

3 = − 3

Using a similar method we determined the values of sin θ, cos θ, and tan θ shown in Table 1.4.

.

EXAMPLE 1: If tanθ = 3

2 and 0< θ < 𝛑

2 , find the five other trigonometric functions

of θ.

Solution : From tanθ = 3

2 we construct the right triangle of height 3 (opposite) and

base 2 (adjacent) in Figure . The Pythagorean theorem gives the length of the

hypotenuse, 4 + 9 = 13. From the triangle we write the values of the other five trigonometric functions:

cos θ = 2

13 sin θ =

3

13

sec θ = 13

2 csc θ =

13

3

cot θ = 2

3

•Periodicity and Graphs of the Trigonometric Functions Periodic Function: A function ƒ(x) is periodic if there is a positive number p such that f(x+p) = f(x) for every value of x. The smallest such value of p is the period of ƒ.

”graph trigonometric functions in the coordinate plane: See Figure,

”As we can see in Figure , the tangent and cotangent functions have period p = 𝛑The other four functions have period 2𝛑.

•Even and Odd Trigonometric functions: The symmetries in the Figure graphs functions , the cosine and secant functions are even and the other four functions are odd:

•Identities:

The coordinates of any point P(x, y) in the plane can be expressed in terms of the point’s distance from the origin and the angle θ that ray OP makes with the positive

x-axis (shown in Figure).

cos θ = 𝑥

𝑟 and sin θ =

𝑦

𝑟 We have

x = r cos θ , y = r sin θ .

When r = 1 we can apply the Pythagorean theorem to the reference right triangle in Figure and obtain the equation .

𝑐𝑜𝑠2 θ + 𝑠𝑖𝑛2 θ = 1 .....................(1)

This equation, Dividing in turn by 𝑐𝑜𝑠2 θ and

𝑠𝑖𝑛2θ gives

1 + 𝑡𝑎𝑛2θ = 𝑠𝑒𝑐2θ.

1 + 𝑐𝑜𝑡2θ = 𝑐𝑠𝑐2θ. ……………………… (1‘)

” Addition Formulas: cos (A + B) = cos A cos B ₋ sin A sin B

sin( A + B) = sin A cos B + cos A sin B

There are similar formulas for .................................(2)

cos (A – B) = cos A cos B + sin A sin B

sin(A – B) = sin A cos B - cos A sin B

For example, substituting θ for both A and B in the addition formulas gives.

”Double-Angle Formulas: cos 2θ = 𝑐𝑜𝑠2θ ₋ 𝑠𝑖𝑛2θ

sin 2θ = 2 sin θ cos θ .............(3)

Additional formulas come from combining the equations

cos2 θ + sin2 θ = 1 ,

𝑐𝑜𝑠2θ ₋ 𝑠𝑖𝑛2θ = cos 2θ.

We add the two equations to get 2𝑐𝑜𝑠2θ = 1 + cos 2θ,

and subtract the second from the first to get 2𝑠𝑖𝑛2θ = 1 – cos 2θ, This results in the

following identities,

” Half-Angle Formulas:

𝒄𝒐𝒔𝟐θ = 1+𝑐𝑜𝑠2θ

2 .......................(4)

𝑠𝑖𝑛2θ = 1 −𝑐𝑜𝑠2θ

2 ........................(5)

”The Law of Cosines: If a, b, and c are sides of a triangle ABC and if u is the angle opposite c, then

𝑐2= 𝑎2+ 𝑏2- 2ab cos θ. ........................(6)

This equation is called the law of cosines.

As in Figure The coordinates of A are (b, 0); the coordinates of B are (a cos θ, a sin θ). The square of the distance between A and B is therefore ,

𝐶2 = (acos θ − 𝑏) 2 + (𝑎𝑠𝑖𝑛θ)2

𝐶2 = 𝑎2(𝑐𝑜𝑠2θ + 𝑠𝑖𝑛2θ) + 𝑏2 − 2abcosθ

When 𝑐𝑜𝑠2θ + 𝑠𝑖𝑛2θ = 1 than,

𝐶2 = 𝑎2+ 𝑏2 − 2abcosθ

EXERCISES 1.7: 1. One of sin x, cos x, and tan x is given. Find the other two if x lies in the specified

interval. a. cos x = - 5

13 , x ∈ [

𝜋

2 , 𝛑〕 b. sin x = -

1

2 x ∈ [𝛑,

3𝛑

2 〕

c. tan x = 1

2 , x ∈ [ 𝛑 ,

3𝛑

2 〕

2. Graph the following functions. What is the period of each function ?

a. cos𝛑𝑥

2 b. ₋ sin

𝛑𝑥

3 c. cos(𝒙 −

𝛑

𝟐 ) d. sin(𝑥 −

𝛑

4 ) + 1

e. S = - tan 𝛑t h. S = sec(𝛑𝑡

2 )

3. Use the addition formulas to derive the identities in the following Exercises.

a. cos(𝒙 −𝛑

𝟐 ) = sin x b. sin(𝒙 −

𝛑

𝟐 ) = − cos 𝑥 𝒄. sin(2𝛑 − x)

e. Evaluate cos11𝛑

12 as cos(

𝛑

4 +

2𝛑

3 ) h. Evaluate sin

5𝛑

12

4. Using the Double-Angle Formulas, Find the function values in following Exercises .

a. 𝒄𝒐𝒔𝟐 𝛑

𝟏𝟐 b. 𝑠𝑖𝑛2 𝛑

8

5.Derive the formula. tan(A + B) = tan A + tan B

1 − tan A tan B ,and derive the formula tan (A – B).

6.What happens if you take B =A in the identity cos(A – B) = cos A cos B + sin A sinB?

7. Apply the law of cosines to the triangle in the accompanying figure to derive the formula for cos(A – B) . 8. Apply the formula for cos (A – B) to the identity sin 𝛉 =

cos 𝜋

2 − 𝜃 to obtain the addition formula for sin(A + B) .

9. A triangle has sides a = 2 and b = 3 and angle C = 60°. Find the length of side c. 10. The law of sines says that if a, b, and c are the sides opposite the angles A, B, and C in a

triangle, then sin A

𝒂 =

sin B𝑏

= sin C

𝑐

Use the accompanying figures and the identity sin ( 𝛑 - 𝛉 ) = sin 𝛉. if required, to derive the law.

11. A triangle has side c = 2 and angles A = 𝛑

4 and B =

𝛑

3 .

Find the length a of the side opposite A.