Download Section 4.1 – Special Right Triangles and Trigonometric Ratios 1

Section 4.1
Special Triangles and Trigonometric Ratios
Some facts about the angles of triangles:
The sum of the three angles of a triangle add up to 180 o .
An acute angle of a triangle is an angle between 0 o and 90 o . A right angle is a 90 o angle.
Complementary angles add up to 90 o .
Supplementary angles add up to 180 o .
In this section, we’ll work with some special triangles before moving on to defining the six
trigonometric functions.
If we are given a right triangle and we’re interested in finding a missing side, we can apply
Pythagorean’s Theorem, a 2 + b 2 = c 2 .
Leg
a
c Hypotenuse
Leg b
Example 1: In the right triangle below, find the missing leg.
4 5 cm
4 cm
Section 4.1 – Special Right Triangles and Trigonometric Ratios
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30-60-90 Triangles
45-45-90 Triangles
Example 2: Find the missing sides.
a.
a
c
b.
122 5
a
30 o
4 2
a
Section 4.1 – Special Right Triangles and Trigonometric Ratios
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The Six Trigonometric Ratios of an Angle
The word trigonometry comes from two Greek roots, trignon, meaning “having three sides,” and
meter, meaning “measure.”
A trigonometric function is a ratio of the lengths of the sides of a triangle. If we fix an angle,
then as to that angle, there are three sides, the adjacent side, the opposite side, and the
hypotenuse. We have six different combinations of these three sides, so there are a total of six
trigonometric functions. The inputs for the trigonometric functions are angles and the outputs
are real numbers.
Let θ be an acute angle places in a right triangle; then
Side
opposite to
angle θ
Hypotenuse
Side adjacent to angle θ
θ
The names of the six trigonometric functions, along with their abbreviations, are as follows:
Name of Function
Abbreviation
cosine
cos
sine
sin
tangent
tan
secant
sec
cosecant
csc
cotangent
cot
length of side adjacent to angle θ
length of hypotenuse
length of side opposite to angle θ
sin θ =
length of hypotenuse
length of side opposite to angle θ
tan θ =
length of side adjacent to angle θ
cos θ =
adjacent
hypotenuse
opposite
sin θ =
hypotenuse
opposite
tan θ =
adjacent
cos θ =
A useful mnemonic device:
SOH-CAH-TOA
S=
O
H
sec θ =
length of hypotenuse
length of side adjacent to angle θ
sec θ =
hypotenuse
adjacent
C=
A
H
csc θ =
length of hypotenuse
length of side opposite to angle θ
csc θ =
hypotenuse
opposite
T=
O
A
cot θ =
length of side adjacent to angle θ
length of side opposite to angle θ
cot θ =
adjacent
opposite
Section 4.1 – Special Right Triangles and Trigonometric Ratios
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Example 3: Suppose a triangle ABC has C = 90o, AC = 7 and AB = 9. Find the values of all six
trigonometric ratios for angle B.
Example 4: Suppose that θ is an acute angle in a right triangle and sec θ =
and cot θ .
Section 4.1 – Special Right Triangles and Trigonometric Ratios
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. Find sin θ
4
4
Example 5: In triangle MNP, angle M is 30 o and angle N is the right angle. State whether each
of the following statements is true or false.
I. sin 30o = cos 60o
II.
TRUE or FALSE
cos30o
= tan 30o TRUE or FALSE
o
sin 30
Try this one: Given 45-45-90, state whether each of the following statements is true or false.
I. sec 45o = csc 45o
II.
TRUE or FALSE
1
= sec 45o TRUE or FALSE
o
sin 45
Section 4.1 – Special Right Triangles and Trigonometric Ratios
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