Download Trigonometric Ratios

LESSON 3: TRIGONOMETRIC RATIOS
Learning Outcomes:

To relate the trigonometric ratios to the coordinates of points in the unit circle

To determine exact values for trigonometric ratios

To identify the measures of angles that generate specific trigonometric values

To solve problems using trigonometric ratios
Recall:
Suppose θ is any angle in standard position, and P(x, y) is any point on its terminal arm, at a
distance r from the origin.
If given the values (x, y) of the two smaller legs of a right
angled triangle, what formula would you use to find the r
value?
Pythagorean Theorem, 𝑟 = √𝑥 2 + 𝑦 2
You can use a reference triangle to determine the three primary trigonometric ratios in terms of
x, y and r.
Knowing that:
Express the sin, cos and tan ratios in terms of x, y and r:
With a unit circle with radius of 1:
𝑥
= 𝑥 , 𝑡ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒
1
𝑦
sin 𝜃 = = 𝑦 , 𝑡ℎ𝑒 𝑠𝑒𝑐𝑜𝑛𝑑 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒
1
cos 𝜃 =
Therefore, you can describe the coordinates of any point 𝑃(𝜃) as (cos 𝜃, sin 𝜃). This is true for
any point 𝑃(𝜃) at the intersection of the terminal arm of an angle θ and the unit circle.
Reciprocal Trigonometric Ratios:
1
cosecant ratio  csc 
sin 
cotangent ratio  cot  
secant ratio  sec 
1
cos
1
tan 
These are reciprocals of the primary trigonometric ratios.
Ex. The point (15, 8) lies on the terminal arm of  as shown. Calculate the
value of r and the exact values of the primary and reciprocal ratios.
1 2√2
Ex. The point B(− 3 ,
3
) lies at the intersection of the unit circle and the terminal arm of an
angle θ in standard position.
a. In what quadrant does point B lie?
b. Determine the values of the six trigonometric ratios for θ. Express your answers in
lowest terms.
Exact values for the trigonometric ratios can be determined using special triangles.
With the knowledge of the special triangles, the CAST rule can be used to tell us where each
trigonometric ratio is positive.
CAST Rule:
Sine ratio
All ratios
The reciprocal trigonometric ratios
positive
positive
follow the same framework as their
corresponding primary ratio.
Tangent
Cosine ratio
ratio
positive
positive
The trigonometric ratios of any angle can be written as the same function of a positive acute
angle called the reference angle with the sign of the ratio being determined by the CAST rule.
Ex. Determine the exact value for each.
a. 210
b. cos
5
3
𝜋
c. 𝑡𝑎𝑛 2
d. sec 60°
e. sin(-300˚)
Assignment: pg. 201-205 #1, 3-6, 8