Transcript 4.4 Day One Trigonometric Functions of Any Angle

4.4
Day 1
Trigonometric Functions of Any Angle
– Use the definitions of trigonometric functions of any angle
– Use the signs of the trigonometric functions
Pg. 499 #2-22 (even)
At any point on the circle, we can connect a vertical line to the x-axis
and create a triangle.
Horizontal side = x, vertical side=y, and hypotenuse=r.
x and y may be positive or negative (depending on their direction)
The radius, r, is always a positive value.
For any point (x,y) found on the circle,
sin Θ = y
cos Θ = x
tan( ) 
sin( ) y

cos( ) x
csc  ) 
1
1
 = Reciprocal of Sine
sin( ) y
sec( ) 
1
1
 = Reciprocal of Cosine
cos( ) x
cot( ) 
cos( ) x

sin( ) y
Given a point (-3,-4), find the 6 trig. functions associated with the angle formed
by the ray containing this point.
x=-3, y=-4, and so r =
Then,
sin A = -4/5, cos A = -3/5, tan A = 4/3
csc A =
Evaluate, if possible, the cosine function at the following four quadrant angles:
2. 0o
3. 90o
4. 180o
5. 270o
Evaluate, if possible, the cosecant function at the following four quadrant angles:
6. 0o
7. 90o
8. 180o
9. 270o
Here’s a helpful anagram to help you remember the signs (positive or negative) of
trigonometric functions:
All
All functions
Snowflakes
Sine and cosecant
Taste
Tangent and cotangent
Cold
Cosine and secant
are (+) in
Quadrant I
are (+) in
Quadrant II
are (+) in
Quadrant III
are (+) in
Quadrant IV
Another way to think of the sign of a function is to remember the variable it is
defined by. Since the radius is always positive, only the signs of x and y influence
the sign of the function.
In any given quadrant:
cosine and secant have the same sign as x
sine and cosecant have the same sign as y
tangent and cotangent have the same sign as the ratio of x and y
Determine the quadrant in which angle Θ lies.
10. sin Θ < 0 and cos Θ > 0
11. sin Θ < 0 and tan Θ > 0
12. sec Θ > 0 and cot Θ < 0
13. tan Θ < 0 and csc Θ < 0