Download Trig – Section 1 - Chipola College

Chipola College
Mac 1114
1.3 and 1.4a__________ ____________________________________________________________
Given that  ABC   XYZ and the measurements shown, fill in the blanks below.
Y
B
A
c
5
12
C
Z
X
c = _______
If YZ = 10, then XZ = ______ and XY = _______
BC

AB
YZ

XY
But since the triangles are similar, we know that m  A = m  X
In relation to angle A (or angle X), THE ratio of the side opposite angle A to the hypotenuse (in EITHER
triangle) is ______.
This value is called the sine of angle A. The definition is given as sin (A) = sin (X) =
opposite
hypotenuse
In comparing the ratios of the other sides, the definitions of the six trigonometric values and their abbreviations
for a given angle are:
sine (A) = sin(A) =
opposite
hypotenuse
cosecant (A) = csc(A) =
tangent (A) = tan (A) =
hypotenuse
opposite
opposite
adjacent
cosine (A) = cos(A) =
secant (A) = sec(A) =
adjacent
hypotenuse
hypotenuse
adjacent
cotangent (A) = cot(A) =
adjacent
opposite
Important Note: In any triangle similar to triangle ABC, the ratios of the sides will
be the same. Thus, the value of THE sin(A) is a constant value.
The name of the Indian Princess “SOH CAH TOA” is useful in remembering these
definitions. Sin = Opp/Hyp, Cos = Adj/Hyp, Tan = Opp/Adj, then use reciprocals
for the csc, sec and cot.
Given the following triangle, determine the six trig functions for angle B.
1) If BC = 3 and AC = 4,
B
A
sin(B) =
cos(B) =
tan(B) =
sec(B) =
cot(B) =
c
a
csc(B) =
12
b
C
2) If BC = 5 and AC = 6,
sin(B) =
cos(B) =
tan(B) =
csc(B) =
sec(B) =
cot(B) =
Adapting these definitions to angles in standard position on the coordinate plane:
1) Estimate the location of the point with coordinates (x, y) in quadrant I.
2) Draw the angle in standard position that passes through the point (x, y).
3) Draw a segment from the point (x, y) perpendicular to the x-axis.
4) You should now have a triangle with one angle at the origin and a right angle at coordinates (___, ____)
5) Label the angles with the vertex at the origin as angle A.
6) What is the length of the side opposite angle A? ___________
7) What is the length of the side adjacent to angle A? ___________
8) Give an expression for the length of the hypotenuse. ______________
9) We call this length r (rather than c).
10) Define the six trig functions for angle A using x, y and r.
sin(A) =
cos(A) =
tan(A) =
csc(A) =
sec(A) =
cot(A) =
These are the definitions for the six trigonometric functions when the angle is in
standard position. Realizing that in each quadrant the values for x and y can be positive
or negative, determine the “sign” of each of the six functions in each quadrant.
Function
Sin (A)
Cos (A)
Tan (A)
Sec (A)
Csc (A)
Cot (A)
Q1
Terminal side of angle A lies in quadrant:
Q2
Q3
Q4
Practice:
Given the ordered pair, determine the six trig functions for an angle whose terminal side passes through the
given point.
1) (5, 8)
2) (-3, 7)
Given one trig function for an angle, determine the other five.
3) tan (  ) =
3
,  in quadrant 3
4
2) cos (  ) = 
2
,  in quadrant 2
5
Special angles – “Quandrantal” angles
Draw each of the following angles in standard position:
90º, 180º, 270º, 360º
Let r have a value of 1, and determine the value of x and y, if (x, y) is on the terminal side of each angle.
Function
90º
Quadrantal Angle A
180º
270º
360º
Sin (A)
Cos (A)
Tan (A)
Sec (A)
Csc (A)
Cot (A)
HW/ See Mac 1114 Unit One Homework Checklist