Download (some of) 4.8: Right Triangle Trigonometry

4.3 & 4.8 Right Triangle Trigonometry
Anatomy of Right Triangles
The right triangle shown at the right uses
lower case a, b and c for its sides with
c being the hypotenuse. The sides a and b
are referred to as the legs of the right triangle.
Upper case A, B and C are its angles
with C being the right angle.
A + B + C = 180° (True for all triangles)
A and B are complementary angles ( A + B = 90° )
Pythagorean Theorem:
a2 + b2 = c 2 ... and all of its rearrangements
c = a2 + b2 (can’t simplify to c = a + b ... TRAGIC mistake!)
a = c 2 − b2
b = c 2 − a2
Handy Pythagorean Triples: just keep in mind when using Pythagorean
triples that the hypotenuse MUST be the longest side of the right triangle.
Similar Right Triangles
Two right triangles are similar if their matching sides are in the same proportion.
The measure of each angle (using a protractor):
α≈
‘alpha’
β≈
‘beta’
θ≈
‘theta’
The trigonometric functions are based on this premise: the ratio of any two sides
of similar right triangles can be paired with the angle measures they share.
These ratios are a function of the angle ... NOT a function of the triangle!
There are 6 such ratios you can create using the sides of a right triangle.
THE SIX TRIGONOMETRIC RATIOS
ADJ* = adjacent leg to angle θ
OPP* = opposite leg to angle θ
HYP = hypotenuse
(* The role of ADJ and OPP depend on which
of the two acute angles is being used)
The ‘main’ three:
sin(θ ) =
OPP
HYP
cos(θ ) =
Pronounced " Sine theta "
ADJ
HYP
"Cosine theta "
tan(θ ) =
OPP
ADJ
"Tangent theta "
... and their reciprocals:
csc(θ ) =
HYP
OPP
"Cosecant theta "
sec(θ ) =
HYP
ADJ
" Secant theta "
cot(θ ) =
ADJ
OPP
"Cotangent theta"
A handy way to remember the main three ratios is to use SOH‐CAH‐TOA.
“ S O H ” stands for “ Sine = O / H ”
“ C A H ” stands for “Cosine = A / H ”
“ T O A ” stands for “Tangent = O / A ”
The reciprocals need to be committed to memory. For sine and cosine’s
reciprocals always remember to pair an ‘S’ with a ‘C’:
Sine’s reciprocal is Cosecant
Cosine’s reciprocal is Secant
It should be easy to remember that tangent and cotangent are reciprocals.
ex) Evaluate all 6 trigonometric ratios for the angle θ shown in the diagram.
(You’ll need to determine the missing value first.)
sin(θ ) =
csc(θ ) =
cos(θ ) =
sec(θ ) =
tan(θ ) =
cot(θ ) =
ex) Evaluate all 6 trigonometric ratios for the angle θ shown in the diagram.
ALWAYS MAKE SURE YOU RATIONALIZE DENOMINATORS!
sin(θ ) =
csc(θ ) =
cos(θ ) =
sec(θ ) =
tan(θ ) =
cot(θ ) =
What does the ‘CO‐’ stand for?
In the triangle shown here, remember that angles α and β are complementary.
Evaluate
sin(α ) =
and
cos(β ) =
Evaluate
tan(α ) =
and
cot(β ) =
Evaluate
sec(α ) =
and
csc(β ) =
For any pair of complementary angles, a trig ratio and its co‐function counterpart
have the same value.
ex) The value of sin(28°) is equivalent to the cos( _____ )
The value of cot(61°) is equivalent to the tan( _____ )
Solving Right Triangle Problems (with a calculator)
The main three trig ratios are built into your calculator.
MAKE SURE YOUR CALCULATOR IS IN THE CORRECT ANGLE MODE!
ex) Evaluate
cos(28 ° )
and
cos(28)
To ‘solve’ a right triangle means to provide all the missing measurements. You’ll
need to take the given information and relate a trigonometric value of an angle to
a ratio of a known and unknown side.
ex) Solve the triangle. Round the side measurements to 2 decimal places.
ex) A service ramp makes an angle of elevation measuring 10.75° . The ramp
needs to accommodate a vertical rise of 1 meter. How long should the ramp be?
Round to the nearest tenth of a meter.
ex) A ship leaves harbor at a bearing of S 75° E at a speed of 50 mph. After 2
hours without changing course, how far south of the harbor has the boat traveled?
Round to the nearest tenth of a mile.
‐1
‐1
What are SIN , COS and TAN
‐1
?
The second functions above SIN, COS and TAN buttons are for determining angles
based on a known sine, cosine or tangent ratio.
ex) If cos(θ ) = 0.32
What is the value of
θ in degrees?
What is the value of
θ in radians?
ex) Determine the angle
shown here.
Special Angle Right Triangles ... YOU NEED TO KNOW THESE!
For 30° or
For 45° or
π
6
π
4
For 60° or
These triangles can take any size BUT
the sides are always proportional to these
values.
For example: for a 30‐60‐90 triangle, the
ratio of the short leg to the hypotenuse
will ALWAYS be 1:2
π
3
ex) Evaluate the following WITHOUT using your calculator.
sin(30 ° )
tan(45 ° )
sec( π /6)
sin( π /4)
sec(60 ° )
tan( π /6)
csc(45 ° )
cos( π /3)
sin( π /4)
cos(30 ° )
cot(60 ° )
csc( π /3)