Name:_________________________________________
Trigonometric Laws – Notes Packet
Date:_______ Period:______
Ms. Anderle
Law of Cosines
Since the Pythagorean Theorem can only be used in right triangle problems. New formulas need
to be learned that can be used when the triangle is not right.
Law of Cosines: ____________________________________________
We can notice that <C and side c are at __________________________ ends of the formula. In
addition, the formula resembles __________________________________________________.
We can rewrite the Law of Cosines for each angle of a triangle. Notice how each pattern of
letters remain the same.
The law of cosines can be used when a problem refers to _____ sides and ______ angle.
In addition, we can use the law of cosines to find a missing ________ or a missing __________.
Examples:
4)
5)
6) In ΔXYZ, if x = 8, y = 9, and m<Z = 135°, find z to the nearest unit.
7) Two sides of a parallelogram form an angle of 72o. The lengths of the two sides are 18 and
24 centimeters. How long, to the nearest centimeter, is the shorter diagonal?
8) In isosceles ΔQRS, q = s = 4.7 cm. If cos R = .1908, find the length of side r to the nearest
hundredth of a centimeter.
9) A surveyor at point R sights two points S and T on opposite sides of a lake. Point R is 120m
from S and 180m from T, and the measure of <R is 38o. Find the distance across the lake to the
nearest meter.
10) A triangular walking course has 2 sides of 230 feet and 360 feet, and the angle between
these sides measures 38°. Find the length of the third side of the course, to the nearest foot.
11) In ΔABC, a = 6, b = 7, and c = 4. Find m<A to the nearest degree.
What is different about this question?
New Formulas to Use:
12) In ΔABC, a = 6, b = 7, and c = 4. Find the largest angle to the nearest degree.
13) In ΔABC, a = 17.5, b = 16.4, and c = 11.7. Find m<C to the nearest tenth.
Summary: When do we use law of cosines?
1)
2)
Law of Sines
Do Now:
1) Find b to the nearest unit
Applying Trigonometry in Working in Triangles Which Do Not Contain a Right Angle:
Notice: -Side a is opposite ______________________
-Side b is opposite ______________________
-Side c is opposite ______________________
In the law of sines, the ratio of each side to the sine of its “partner” are equal to each other.
***If a problem refers to ____ sides and ____ angles, use the Law of Sines***
LAW OF SINES:
These ratios, in pairs, are applied to solving problems. You never need to use all three ratios
at the same time. Mix and match the ratios to correspond with the letters you need.
Remember when working with proportions, the product of the means equals the product of
the extremes (cross multiply).
Examples:
2) In ΔABC, m<A = 42o10', m<B = 73o18', b = 36.14. Find a to the nearest tenth.
3) In ΔABC, a = 80, c = 60, and m<A = 52o. Find m<C to the nearest degree.
4) Two angles of a triangle measure 40o and 56o. If the longest side is 28, find the length of the
shortest side, to the nearest unit.
5) In ΔABC, a = 16.3, m<B = 58o, m<C = 37o. Find m<A and b and c to the nearest tenth.
6) In an isosceles triangle, the vertex angle is 30º and the base measures 12 cm. Find the
perimeter of the triangle to the nearest integer.
Summary: When do we use the law of sines?
1)
2)
Ambiguous Case:
Law of Sines
In Geometry, we found that we could prove triangle congruent using: _____________________
________________________________. We also discovered that we cannot use ________ to
prove triangle congruent.
By definition, the word ambiguous means open to two or more interpretations. Such is the case
for certain solutions when working with the law of sines.
If you are given two angles and one side (AAS or ASA) the law of sines will work nicely and give
you back one side.
However, if you are given two sides and one angle, then the law of sines could possibly provide
you with one solution, two solutions, or no solutions.
Facts that we must remember when using the law of sines:
1) ________________________________________________________________________
________________________________________________________________________
2) ________________________________________________________________________
3) ________________________________________________________________________
4) ________________________________________________________________________
________________________________________________________________________
Steps to Using the Ambiguous Case:
1)
2)
3)
Examples:
How many distinct triangles can be drawn given these measurements?
1) 1. a = 7, b = 10, m<A = 30o
2) m<A = 140o, b = 10, a = 3
3) m<A = 50o, a = 13, c = 16
4) m<C = 45o, c = 8, b = 8 2
5) m<B = 45o, a = 20 2 , b = 18
6) m<C = 110o, c = 8, a = 4
Area of a Triangle and Parallelogram Using Trigonometry
We are all familiar with the formula for area of a triangle: ______________________.
Using right triangle trigonometry, we can come up with a new formula for area of a triangle!!!
New Formula:
The “letters” of the formula may change with the problem, so we need to remember the patter
of ___________________________________________________________.
Examples:
1) In ΔABC, a = 8, b = 12, and m<C = 140o. Find the area to the nearest square unit.
2) In an isosceles Δ, the two equal sides each measure 24 meters, and they include an
angle of 30º. Find the area of the isosceles triangle, to the nearest sq. meter.
3) In ΔABC, b = 10, c = 8, and m<A = 60o. Find the EXACT area.
4) In ΔPQR, q = 15 and r = 60. Find the measure of <P, to the nearest degree, if the area is
240 square units.
5) The lengths of two sides of a parallelogram are 24 and 36 cm. Their included angle
measures 50o. Find the area of the parallelogram to the nearest tenth.
Area of Parallelogram:
6) A farmer has a triangular field where two sides measure 450 yards and 320 yards. The
angle between these two sides measures 80º. The farmer wishes to use an insecticide
that costs $4.50 per 100 sq. yards or any part of 100 yds. What will it cost to use this
insecticide on this field?
7) The Art Guild is painting a mural in the shape of an isosceles triangle on their building at
the State Fair. The equal sides of the triangle will each be 12 yards in length, and the
area of the triangular mural will be 55 square yards. Find the measure of the three
angles of the triangle, to the nearest hundredth of a degree.