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Trigonometric Functions of Acute
Angles

Let us consider the right triangle ABC,
with right angle at C. Angles A and B are
acute angles which are complementary
(A + B = 90°). The sides opposite angles
A, B, and C will be denoted by the
corresponding small letters a, b, and c,
respectively. Then by taking ratios of the
sides of the triangle, we define the three
trigonometric functions of acute angle A
as follows:
sine A =
cosine A =
tangent A =
𝑠𝑖𝑑𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐴
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑠𝑖𝑑𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 𝐴
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑠𝑖𝑑𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐴
𝑠𝑖𝑑𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 𝐴

By using the abbreviations of the three
trigonometric functions and the
corresponding sides of the triangle with
reference to angle A, the
corresponding equations are:
sine A =
𝑠𝑖𝑑𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐴
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
cosine A =
𝑠𝑖𝑑𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 𝐴
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
tangent A =
𝑠𝑖𝑑𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐴
𝑠𝑖𝑑𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 𝐴
𝑎
sin A =
𝑐
𝑏
cos A =
𝑐
𝑎
tan A =
𝑏

The other three trigonometric functions that
can be derived from triangle ABC are as
follows with the corresponding
abbreviations.
cosecant A =
secant A =
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑠𝑖𝑑𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐴
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑠𝑖𝑑𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 𝐴
cotangent A =
𝑠𝑖𝑑𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 𝐴
𝑠𝑖𝑑𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐴
𝑐
csc A =
𝑎
𝑐
sec A =
𝑏
𝑏
cot A =
𝑎
In summary, acute angles (angle A & angle
B) are the reference angles of the right
triangle:
 We use the symbol θ (Greek letter “theta”)
for these acute angles or capital letters A
or B. Thus, θ = A or θ = B.

𝑎
sin θ = sin A =
𝑐
𝑏
cos θ = cos A =
𝑐
𝑎
tan θ = tan A =
𝑏
𝑐
csc θ = csc A =
𝑎
𝑐
sec θ = sec A =
𝑏
𝑏
cot θ = cot A =
𝑎
You can use mnemonics!
S oh C ah T oa
S oh
C ah
T oa
𝑜𝑝𝑝.
sin θ =
ℎ𝑦𝑝.
𝑎𝑑𝑗.
cos θ =
ℎ𝑦𝑝.
𝑜𝑝𝑝.
tan θ =
𝑎𝑑𝑗.
You can use mnemonics!
C ho S ha C ao
C ho
ℎ𝑦𝑝.
csc θ =
𝑜𝑝𝑝.
S ha
ℎ𝑦𝑝.
sec θ =
𝑎𝑑𝑗.
C ao
𝑎𝑑𝑗.
cot θ =
𝑜𝑝𝑝.

It will be noted that the first three functions
are the reciprocals of the other three, thus
we may write:
1
sin θ =
csc θ
1
cos θ =
sec θ
1
tan θ =
cot θ
1
csc θ =
sin θ
1
sec θ =
cos θ
1
cot θ =
tan θ
Note: The reciprocal of a number is 1 divided
by the number.
Pythagorean Theorem

In a right triangle, to find the value of
side c which is the hypotenuse of
triangle ABC, we may use the
Pythagorean theorem which states that:
the square of the hypotenuse of a right
triangle equals the sum of the squares
of its legs. In symbols: c2 = a2 + b2
Derivations:
c=
a = 𝒄𝟐 −𝒃𝟐
𝒂𝟐 +𝒃𝟐
b = 𝒄𝟐 −𝒂𝟐
Examples
1.
In a right triangle ABC where angle C is
right, and sides a and b have lengths 3
and 4 units, respectively, find the
measure of side c and derive the six
trigonometric functions with reference
to angle A.
12
2. Given sin A =
, A is acute; find the
13
other trigonometric function values of A.
Exercises
1.
2.
3.
Draw the right triangle whose sides are
a = 2, b = 4, and c = 20 and find the
six trigonometric functions of angle B.
Solve for the unknown side using the
Pythagorean theorem given the sides
a = 2 and b = 3.
If tangent A = 5, what are the other
trigonometric function values of A, given
A is acute?
Do Worksheet 3
Trigonometric Functions of Any Angle θ
The trigonometric functions, which we are
now ready to define, are illustrations of
functions whose domains are sets of angles.
 Using the Cartesian Coordinate System,
we will define the six trigonometric ratios.
 Let θ be an angle in standard position.
Choose any point P(x, y) on the terminal side
of θ. Draw a perpendicular line from P to the
x-axis, thus forming a right triangle of
reference for θ.


The point P has three coordinates: the abscissa x, the
ordinate y, and the hypotenuse r in terms of which we
define the following trigonometric functions.
sin θ =
𝒐𝒓𝒅𝒊𝒏𝒂𝒕𝒆
𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆
cos θ =
𝒂𝒃𝒔𝒄𝒊𝒔𝒔𝒂
𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆
tan θ =
𝒐𝒓𝒅𝒊𝒏𝒂𝒕𝒆
𝒂𝒃𝒔𝒄𝒊𝒔𝒔𝒂
csc θ =
𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆
𝒐𝒓𝒅𝒊𝒏𝒂𝒕𝒆
sec θ =
𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆
𝒂𝒃𝒔𝒄𝒊𝒔𝒔𝒂
cot θ =
𝒂𝒃𝒔𝒄𝒊𝒔𝒔𝒂
𝒐𝒓𝒅𝒊𝒏𝒂𝒕𝒆
𝒚
=
𝒓
𝒙
=
𝒓
𝒚
=
𝒙
𝒓
=
𝒚
𝒓
=
𝒙
𝒙
=
𝒚
If the coordinates of P(x, y) are given,
we can derive the six trigonometric
functions of angle θ in standard position.
 Since r is the hypotenuse of the right
triangle, we can solve for its value using

the Pythagorean relation: r =
𝒙𝟐 + 𝒚𝟐
Signs of the Six Trigonometric Functions
Note: r is always positive. Thus, the signs of the
values of the trigonometric functions of angle θ
are determined by the signs of x and y.

The signs of the trigonometric functions depend
upon the quadrant in which the terminal side of θ
lies.
A S T C - All Students Take Calculus!
A S T C - All Silly Trig Classes!
Examples
1.
2.
The terminal side of angle θ, in standard
position, passes through the point P(–3, 4).
Draw θ and find the values of the six
trigonometric functions of θ.
Let θ be an angle in standard position. Given
12
cos θ =
. Find the other trigonometric
13
function values if the terminal side is in the
fourth quadrant.
Reference Angle
The trigonometric function values of any
angle θ can be determined by knowing the
function values of acute angles (an angle
π
having positive radian measure less than or
2
less than 90°). The procedure involves the
concept of a reference angle.
 Reference angle – is defined as the smallest
positive acute angle θ associated with a nonquadrantal angle in standard position formed
by the terminal side and the x-axis.

Reference Angles

To find the reference angle θ’ of an angle θ (given angle)
with positive rotation and whose measure is less than 360° or
2π, the following diagrams illustrate the relationship between
θ and θ’ in each of the four quadrants. In this case, θ = A.

The following table summarizes how to find the
reference angle if the given angle θ < 360°.
If the given
angle θ is in
quadrant
Reference angle θ’ is
(Degrees)
(Radians)
I
II
III
θ
180° - θ
θ - 180°
θ
π-θ
θ-π
IV
360° - θ
2π - θ

To find the reference angle for an angle with
negative measures or for an angle greater
than 360°, first find a coterminal angle whose
measure is between 0° and 360°. Then, use
the appropriate formula summarized in the
given table.
Examples
Given θ is an angle in standard position. Find
the measure of its reference angles θ’.
1. 250°
2. 330°
3. 125° 45’
Exercises
I. In which quadrants do tan θ and sin θ have
opposite signs?
II. The terminal side of an angle α in standard
position passes through the point M(-3, -3). Find
the trigonometric functions of α.
III. Given θ is an angle in standard position. Find
the measure of its reference angle.
1.
2.
3.
4.
5π
6
23π
12
–120°
470°
Do Worksheet 4