TRIGONOMETRY
CHAPTER 2
TRIGONOMETRY
What is it and what do we use it
for?
Trigonometry is a branch of mathematics that
studies relationships involving lengths and angles
of triangles.
What is it and what do we use it for?
Trigonometry is commonly used in finding the height of towers and mountains.
It is used in navigation to find
the distance of the shore from
a point in the sea
It is used in oceanography
in calculating the height of
tides in oceans
It is used in finding the
distance between
celestial bodies
NEW VOCABULARY
ACUTE angle
The angle that is less than 90˚
OBTUSE angle
The angle that is more than 90˚
NEW VOCABULARY
Angle of inclination:
The angle of inclination of a line or line
segment is the acute angle it makes
with the horizontal.
NEW VOCABULARY
ADJACENT side to an angle
OPPOSITE side to an angle
Construct Understanding
• Work with a partner.
• You will need grid paper, a ruler, and a protractor.
A. On grid paper, draw a right ∆ABC with ∠B = 90°.
B. Each of you, draw a different right triangle that is similar to
∆ABC. (What are similar triangles?)
C. Measure the sides and angles of each triangle and label your
diagrams with the measures.
D. The two shorter sides of a right triangle are its legs. Calculate
the ratio of the legs as a decimal, then the corresponding
ratio for each of the similar triangles.
E. How do the ratios compare?
F. What do you think the value of each ratio depends on?
The Ratio of Legs = THE TANGENT RATIO
• Length of side opposite ∠A : Length of side adjacent to
∠A
• depends only on the measure of the angle, not on how
large or small the triangle is.
• Usually written as a fraction
THE TANGENT RATIO
• The Tangent Ratio for ∠A is written as tan A
• The Tangent Ratio for ∠C is written as tan C
C
C
C
The Value of THE TANGENT RATIO
• The value of this ratio is usually expressed as a decimal
without a unit
For Example:
o If tan A = 1.5, then in any similar triangle with ∠A, the length
of side opposite to ∠A is 1.5 times greater than the length of
the side adjacent to ∠A
The Value of THE TANGENT RATIO
For Example:
o If tan A = 1.5, then in any similar triangle with ∠A, the length
of side opposite to ∠A is 1.5 times greater than the length of
the side adjacent to ∠A
Determine tan D and tan F.
Opposite
Adjacent
Determine tan D and tan F.
Adjacent
Opposite
POWERPOINT PRACTICE PROBLEM
Determine tan X and tan Z.
The Value of THE TANGENT RATIO
• The value of this ratio can also be used to find the
value of an acute angle
The Value of THE TANGENT RATIO
• The value of this ratio can also be used to find the
value of an acute angle
Determine the measures of ∠G and ∠J to the nearest tenth of a
degree.
Determine the measures of ∠G and ∠J to the nearest tenth of a
degree.
POWERPOINT PRACTICE PROBLEM
Determine the measures of ∠K and ∠N to the nearest tenth of a
degree.
A 10-ft. ladder leans against the side of a building with its base 4 ft. from
the wall. What angle, to the nearest degree, does the ladder make with the
ground?
A 10-ft. ladder leans against the side of a building with its base 4 ft. from
the wall. What angle, to the nearest degree, does the ladder make with the
ground?
• Draw a diagram.
• Label the vertices of the triangle
PQR.
• To use the tangent ratio to
determine ∠R, we first need to
know the length of PQ.
• Use the Pythagorean Theorem in
right ∆PQR.
A 10-ft. ladder leans against the side of a building with its base 4 ft. from
the wall. What angle, to the nearest degree, does the ladder make with the
ground?
PQ² = PR² - QR²
PQ² = 10² - 4²
PQ² = √84
The angle between the ladder and
the ground is approximately 66°.
PQ √84
tan R =
=
QR
4
√84
tan R =
4
√84
-1
∠R = tan (
) = 66.4218° = 66°
4
HOMEWORK
O PAGE: 75 - 77
O PROBLEMS: 3, 4, 5, 12, 17, 20, 21,