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By the end of this chapter, you should be familiar with:
We already know how to use degrees when measuring angles, but most science and engineering applications use radians. Radians are an alternative way of measuring the size of an angle.
A radian is the meausre of the angle with its vertex at the centre of the circle and two radii with their end points on the circumference and these two points are 1 radii apart.
Note that, the circumference of a circle is 2𝜋𝑟 which means there are 2𝜋 radians in a circle. We also know that a full circle has 360°
Therefore, 2𝜋 radians = 360° which means radians = 180°
Conversion between radians and degrees:
Ex. Convert 30° into radians.
30° = 30π/180 radians = π/6 radians
Ex. Convert π/6 into degrees.
π/6 radians = (π/6) + (180 π) degrees = 30°
Degree measure and radian measure for common angles:
For a circle of radius 𝒓 , a central angle 𝜽 subtends an arc of the circle of length s given by , where 𝒔 = 𝒓𝜽, where 𝜃 is measured in radians.
If the angle is in degrees, then we use s = 2πr(𝜽/360)
Ex. A circle has a radius of 10 centimeters. Find the length of the arc of the circle subtended by a central angle of 150°.
s = 2πr(𝜽/360)
s = 2π(10)(150/360)
s = 25π/3
s = 26.18s
We know that the length of an arc of a circle is given by 𝒔 = 𝒓𝜽
The area of of sector will be A = (πr^{2}/2πr)(r𝜽)
A = (1/2)r^{2}𝜽
We consider a circle with centre at , radius of one unit and equation .
Now, Angles in standard position in the unit circle are measured starting on the positive -axis and turning anticlockwise (positive angles) or clockwise (negative angles).
Now, the coordinate axis divides the plane into four quadrants, first, second, third and fourth.
The and -coordinates of the unit circle are used to define the trigonometric ratios sine, cosine and tangent. (sin, cos, tan).
The -coordinate is assigned to the sin function -coordinate is assigned to the cosine function and the ratio of the two coordinates y/x is assigned to the tangent function. tan𝜽 =(sin𝜽/cos𝜽)
Now, the signs of the trigonometric ratios in each quadrant is defined as:
The standard position of an angle is the position of an angle with its vertex at the origin of a unit circle and one side fixed on the positive -axis. This side is called the initial side of the angle. The other side of the angle, called the terminal side, will intersect the circle at a point.
Keeping this idea in mind, the following formulas are derived:
We don’t need to memorise the formulas, just understand them with intuition and generate them when required.
Ex. Evaluate the three trigonometric functions for arc length l = 3π/2
An arc length of l = 3π/2 is equivalent to three-quarters of the cricumference of the unit circle, so it terminates at the point(0,-1) . By definition: sin(3π/2) = y = -1 , cos(3π/2) = x = 0 , sin(3π/2) = y/x = -1/0 is undefined.
Ex. Find another angle less 360° than that has the same cosine as 50°.
Now, some common values of trigonometric ratios to memorise:
Using these values, most of the trigonometric ratios can be evaluated without using GDC.
SINE CURVE
𝑦 = sin x
COSINE CURVE
y=cosx
Cosine curve is obtained by shifting the sine curve horizontally left by π/2 units. All properties of cosine curve is same as that of the sine curve.
TANGENT CURVE
y = tanx
We have already dicsussed transformation of graphs in previous chapters. Let’s take a brief review:
Function notation | Type of transformation |
𝑓(𝑥) + 𝑑 | Vertical translation up units |
𝑓(𝑥) – 𝑑 | Vertical translation down units |
𝑓(𝑥 + c) | Horizontal translation left units |
𝑓(𝑥 – c) | Horizontal translation right units |
-𝑓(𝑥) | Reflection over -axis |
𝑓(-𝑥) | Reflection over -axis |
𝑎𝑓(𝑥) | Vertical stretch for Vertical compression for |
𝑓(b𝑥) | Horizontal compression for Horizontal stretch for |
Ex. Sketch the graph and find the period and amplitude :
Solution:
For most trigonometric equations there are infinitely many values of the variable that satisfy the equation In order to restrict the number of solutions, we are asked for the solution to be contained within a suitable interval. Although it is possible to write a general expression using a parameter that specifies the infinite values that solve the given equation (general solution).
We will only deal with the exact solutions. (an interval will always be given)
Ex. Find the excat solution(s) to the equation 2 cos 𝑥 − 1 = 0 for 0 ≤ 𝑥 ≤ 2𝜋 We have – 2 cos 𝑥 − 1 = 0
2cosx = 1
cosx = 1/2
The value of cosine function is 1/2 at π/3 and 5π/3 between 0 and 2𝜋.
x = π/3, 5π/3
We can solve this by considering unit circle as well:
The value of -coordinate (cosine) 1/2 is π/3 at 5π/3
Ex. Find the solution(s) to the equation 1 + tan 𝑥 = 0 for −𝜋 ≤ 𝑥 < 𝜋
Graph the equation 1 + tan 𝑥 = 0 and find all the 𝑥-intercepts between – 𝜋 and 𝜋
Looking at the graph 𝑥 = 2.36, −0.785
Ex. Solve 3 sin 𝑥 + tan 𝑥 = 0 for 0 ≤ 𝑥 ≤ 2𝜋
We know that tanx = sinx/cosx
3sinx + (sin/cosx) = 0
3sinxcosx + sinx = 0
sinx(3cosx + 1) = 0
sinx = 0
cosx = (-1/3)
for sin 𝑥 = 0, 𝑥 = 0, 𝜋, 2𝜋
We use the graph of y=cosx and y = -(1/3) and locate the intersection points:From the graph 𝑥 = 1.91, 4.37
Therefore the full solution set will be – 𝑥 = 0, 𝜋, 2𝜋, 1.91, 4.37
Pythagorean identity: We know that the unit circle that we consider has the equation:
𝑥^{2} + 𝑦^{2} = 1
The -coordinate is the cosine ratio and the -coordinate is the sine ratio, therefore: sin^{2}x + cos^{2}x = 1
sin^{2}x = 1-cos^{2}x
cos^{2}x = 1-sin^{2}x
Double angle identities:
cos2x = cos^{2}x – sin^{2}x
sin2x = 2 sinxcosx
The double angle identity for cosine can be also be written as: (using pythagorean property)
sin^{2}x = (1-cos2x)/2
cos^{2}x = (1+ cosx)/2
Ex. Solve cos2x + sin^{2}x = 1/2 ,0 < x < π/2 Using double angle identity for cosine: cos^{2}x – sin^{2}x + sin^{2}x = 1/2
cos^{2}x =1/2
cos 𝑥 = ±1/√2
But we are given the interval only for the first quadrant, therefore: x = π/4
Ex. Solve -2cos^{2}x -sinx + 1 = 0 , 0 ≤ 𝑥 ≤ 4π,
-2cos^{2}x – sinx + 1 = 0 Using pythagorean property to replace cos^{2}x:
-2(1-sin^{2}x)- sinx +1 = 0
2sin^{2}x – sinx -1 = 0
Factorising:
(2sinx + 1)(sinx -1) =0
sinx = -(1/2), sinx = 1
x = 7π/6 , 11π/6
These are the angles from the unit circle, we will add 2𝜋 to each to get all solutions from 0 to 4𝜋.
x =π/2, 7π/6, 11π/6, 5π/2, 19π/6, 23π/6
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