MATHEMATICS: FORM FOUR: Topic 5 - TRIGONOMETRY
Thursday, July 19, 2018
MATHEMATICS: FORM FOUR: Topic 5 - TRIGONOMETRY
July 19, 2018
Trigonometry is a branch of mathematics that deals with relationship (s) between angles and sides
The Sine, Cosine and Tangent of an Angle Measured in the Clockwise and Anticlockwise Directions
Determine the sine, cosine and tangent of an angle measured in the clockwise and anticlockwise directions
The basic three trigonometrical ratios are sine, cosine and tangent which are written in short as Sin, Cos, and tan respectively.
Consider the following right angled triangle.
Also we can define the above triangle ratios by using a unit Circle centered at the origin.
If θis an obtuse angle (90
) then the trigonometrical ratios are the same as the trigonometrical ratio of 180
The angles are as given below
If θ is reflex angle (180<θ<270) then the trigonometric ratios are the same as that of θ-180
If θis a reflex angle (270
< θ< 360
), then the trigonometrical ratios are the same as that of 360
We have seen that trigonometrical ratios are positive or negative depending on the size of the angle and the quadrant in which it is found.
The result can be summarized by using the following diagram.
Trigonometric Ratios to Solve Problems in Daily Life
Apply trigonometric ratios to solve problems in daily life
Write the signs of the following ratios
is in the second quadrant, then Sin 170
= Sin (180
) = Sin 10
= Sin 10
b) Cos 240
= -Cos (240
= -Cos 60
Therefore Cos 240
= -Cos 60
c) Tan 310
= -Tan (360
) = - Tan 50
Therefore Tan 310
= -Tan 50
d) Sin 300
= -sin (360
) = -sin 60
Therefore sin 300
= - Sin 60
Relationship between Trigonometrical ratios
The above relationship shows that the Sine of angle is equal to the cosine of its complement.
Also from the triangle ABC above
Again using the ΔABC
Given thatA is an acute angle and Cos A= 0.8, find
If A and B are complementary angles,
If A and B are complementary angle
Then Sin A = Cos B and Sin B = Cos A
Given that θand βare acute angles such that θ+ β= 90
and Sinθ= 0.6, find tanβ
Sine and Cosine Functions
Sines and Cosines of Angles 0 Such That -720°≤ᶿ≥ 720°
Find sines and cosines of angles 0 such that -720°≤ᶿ≥ 720°
Positive and Negative angles
An angle can be either positive or negative.
: is an angle measures in anticlockwise direction from the positive X- axis
: is an angle measured in clockwise direction from the positive X-axis
From the above figure if is a positive angle then the corresponding negative angle to is (- 360
) or (+ - 360
.If is a negative angle, its corresponding positive angle is (360+)
Find thecorresponding negative angle to the angle θif ;
What is the positive angle corresponding to - 46°?
The angles included in this group are 0
, and 360
Because the angle 0
, and 360
, lie on the axes then theirtrigonometrical ratios are summarized in the following table.
The ∆ ABC is an equilateral triangle of side 2 units
For the angle 45
consider the following triangle
The following table summarizes the Cosine, Sine, and tangent of the angle 30
The following figure is helpful to remember the trigonometrical ratios of special angles from 0°to 90°
If we need the sines of the above given angles for examples, we only need to take the square root of the number below the given angle and then the result is divided by 2.
Find the sine,cosine and tangents of each of the following angles
Find the value of θif Cos θ= -½ and θ≤ θ≤ 360°
Since Cos θis – (ve), then θlies in either the second or third quadrants,
Now - Cos (180 –θ= - Cos (θ+180
) = -½= -Cos60
So θ= 180
or θ= 180
Solve the Following.
The Graphs of Sine and Cosine
Draw the graphs of sine and cosine
Consider the following table of value for y=sinθ where θranges from - 360°to 360°
For cosine consider the following table of values
From the graphs for the two functions a reader can notice that sinθand cosθboth lie in the interval -1 and 1 inclusively, that is -1≤sinθ1 and -1≤cosθ≤1 for all values of θ.
The graph of y= tanθis left for the reader as an exercise
the symbol ∞
Also you can observe that both Sinθnd cosθrepeat themselves at the interval of
360°, which means sinθ= sin(θ+360) = sin(θ+2x360
Each of these functions is called a period function with a period 360
1. Usingtrigonometrical graphs in the interval -360
Find θsuch that
Use the graph of sinθto find the value ofθif
4Sinθ= -1.8 and -360
Sinθ= -1.8÷4 = -0.45
So θ= -153
The graphs of sine and cosine functions
Interpret the graphs of sine and cosine functions
Use thetrigonometrical function graphs for sine and cosine to find the value of
)= - 0.64
Sine and Cosine Rules
The Sine and Cosine Rules
Derive the sine and cosine rules
Consider the triangle ABC drawn on a coordinate plane
From the figure above the coordinates of A, B and C are (0, 0), (c, 0) and(bCosθ, bSinθ) respectively.
Now by using the distance formula
Consider the triangle ABC below
From the figure above,
Note that this rule can be started as “In any triangle the side are proportional to the Sines of the opposite angles”
The Sine and Cosine Rules in Solving Problems on Triangles
Apply the sine and cosine rules in solving problems on triangles
Find the unknown side and angle in a triangle ABC given that
c= 8.6cm and C= 80°
Find the unknown sides and angle in a triangle ABC in which a= 22.2cmB= 86°and A= 26°
By sine rule
Sin A= sin B= Sin C
Find unknown sides and angles in triangle ABC
Where a=3cm, c= 4cm and B= 30°
By cosine rule,
Find the unknown angles in the following triangle
1. Given thata=11cm, b=14cm and c=21cm, Find the Largest angle of ΔABC
2. If ABCD is a parallelogram whose sides are 12cm and 16cm what is the length of the diagonal AC if angle B=119°?
3. A and B are two ports on a straight Coast line such that B is 53km east of A. A ship starting from A sails 40km to a point C in a direction E65°N. Find:
The distance a of the ship from B
The distance of the ship from the coast line.
4. Find the unknown angles and sides in the following triangle.
5. <!--[endif]-->A rhombus has sides of length 16cm and one of its diagonals is 19cm long. Find the angles of the rhombus.
The Compound of Angle Formulae or Sine, Cosine and Tangent in Solving Trigonometric Problems
Apply the compound of angle formulae or sine, cosine and tangent in solving trigonometric problems
The aim is to express Sin (α±β) and Cos (α±β) in terms of Sinα, Sinβ, Cosαand Cosβ
Consider the following diagram:
From the figure above <BAD=αand <ABC=βthus<BCD=α+β
For Cos(α±β) Consider the following unit circle with points P and Q on it such that OP,makes angleα with positive x-axis and OQ makes angle βwith positive x-axes.
From the figure above the distance d is given by
1. Withoutusing tables find the value of each of the following:
1. Withoutusing tables, find:
2. FindSin 225° from (180°+45°)
3. <!--[endif]-->Verify that
Sin90° = 1 by using the fact that 90°=45°+45°
Cos90°=0 by using the fact that 90°=30°+60°
4. <!--[endif]-->Express each of the following in terms of sine, cosine and tangent of acute angles.
5. <!--[endif]-->By using the formula for Sin (A-B), show that Sin (90°-C)=Cos C
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