Math 150 – Fall 2015
Section 8B
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Section 8B – Trig Functions
Pythagorean Theorem: If a right triangle has legs of length a and b and hypotenuse
of length c, then
a2 + b2 = c2
Special Triangles: In a
equal.
π π π
4, 4, 2
triangle (45◦ , 45◦ , 90◦ ), the length of the legs are
In a π6 , π3 , π4 triangle (30◦ , 60◦ , 90◦ ), the hypotenuse is twice as long as the length of
the leg across from the π3 triangle.
Definitions of Trig Functions using a Triangle
Definition. For angles between 0 and π2 , we can define the trig functions at θ using
any right triangle containing the angle θ as below.
opp
hyp
adj
cos θ = hyp
tan θ = opp
adj
sin θ =
Remember: “sohcahtoa”
csc θ =
sec θ =
cot θ =
hyp
opp
hyp
adj
adj
hyp
Math 150 – Fall 2015
Section 8B
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Definitions of Trig Functions using a Circle
Definition. We can define the trig functions at θ for any value of θ by interpreting θ
as an angle in standard position and defining the trig functions using the point (x, y)
where the terminal side of the angle intersects a circle at the origin. (Due to similar
triangles we may use a circle of any radius).
If the point is (x, y), and the circle has radius r, then the trig functions are:
sin θ = yr
csc θ = xr
x
cos θ = r
sec θ = xr
y
tan θ = x
cot θ = xy
If we use the unit circle (r = 1), then the point (x, y) = (cos θ, sin θ).
Note. Where is each trig function positive? Remember, All Students Take Calculus
Example 1. Evaluate the following trig functions:
(a) sin π6
(b) cos 5π
3
(c) tan 5π
4
Math 150 – Fall 2015
Section 8B
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(d) cot 2π
3
(e) sec 7π
6
(f) csc 7π
4
(g) sec 3π
2
(h) tan π2
Example 2. In what quadrant does α lie if cos α > 0 and csc α < 0?
Example 3. If tan β =
6
7
and sin β < 0, find the values of all the trig functions.
Example 4. If sin x = 0.8 and x is not an acute angle, what is the value of tan x?
Note. Using the definitions, we have the identities:
sin θ
tan θ = cos
sec θ = cos1 θ
θ
cos θ
cot θ = sin θ
csc θ = sin1 θ .