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Mathematics
Grade 10
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SESSION 7: DEFINING TRIG RATIOS ON THE CARTESIAN PLANE
KEY CONCEPTS:
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Reciprocal of Trig Ratios
The Cartesian plane
CAST diagram
Special angles 0o and 90o
X-PLANATION
Definition of Trig Ratios for triangles:
Defining the Reciprocal Ratios
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For triangle ABC we define the reciprocal ratios as follows:
Trigonometry on the Cartesian plane
A line, OP, drawn from the origin, will form an angle A with the x-axis. We measure
angle A from the x-axis in an anti-clockwise direction.
X
We can construct a right and triangle OPX, with OP as the hypothenuse. Since the
line OP can rotate about O, we think of it as a radius and call its length r. The x coordinates of P, gives the size of the side adjacent to A and the y co-ordinates of P,
gives the size of the side opposite to A. Now we can re-define the trigonometric
ratios on the Cartesian plane as follows:
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The CAST diagram
In the Cartesian plane, we can use our definitions of the trig ratios to deal with
angles greater than 900. However, the sign of the ratios will change in
Quadrants II, III and IV. The CAST diagram is a memory tool to help us
remember which ratios are positive in the different quadrants.
Special Angles
We expand our table of special angles to include 00 and 900
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X-AMPLE QUESTIONS:
Question 1:
P(3; 4) is a point on the Cartesian plane. XOP = θ, where X is a positive point on the
x-axis. Without using a calculator, determine the value of:
a.) cos θ
b.) 3 tan θ
c.) ½ cosec θ
Question 2:
XOK = θ is an angle in the third quadrant and K is the point (-5; y). OK is 13 units.
Determine without using a calculator.
a.)
b.)
The value of y
Prove that tan2 θ + 1 = sec2 θ
Question 3:
If sin θ = 0,4 and θ is an obtuse angle, determine:
a.)
b.)
cos θ
√21 tan θ
Question 4:
Given tan θ =t /2 , where 0o ≤ θ ≤ 90o. Determine the following in terms of t:
a.)
b.)
c.)
d.)
sec θ
cot θ
cos2 θ
tan2 θ - sec2 θ
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