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MATHEMATICS: FORM FOUR: Topic 5 - TRIGONOMETRY
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.
Positive angle: is an angle measures in anticlockwise direction from the positive X- axis
Negative angle: 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 (- 3600) or (+ - 3600)
- .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 00, 300, 450, 600, 900, 1800, 2700, and 3600
Because the angle 00, 900, 1800, 2700, and 3600, 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 450 consider the following triangle
The following table summarizes the Cosine, Sine, and tangent of the angle 300 , 450 and 600
NB: 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+1800) = -½= -Cos600
So θ= 1800-600 = 1200 or θ= 1800 + 600 = 2400
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
NB: -∞≤ tanθ≤∞the symbol ∞means infinite
Also you can observe that both Sinθnd cosθrepeat themselves at the interval of
360°, which means sinθ= sin(θ+360) = sin(θ+2x3600) etc
and Cosθ=(Cosθ+3600)= Cos(θ+2x3600)
Each of these functions is called a period function with a period 3600
1. Usingtrigonometrical graphs in the interval -3600≤θ≤3600
Use the graph of sinθto find the value ofθif
4Sinθ= -1.8 and -3600 ≤θ≤3600
So θ= -1530, -270, 2070, 3330
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
- Sin (-400)= - 0.64
- Cos (-400)= 0.76
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
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
Find the unknown sides and angle in a triangle ABC in which a= 22.2cmB= 86°and A= 26°
Find unknown sides and angles in triangle ABC
Where a=3cm, c= 4cm and B= 30°
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