Download 3rd Quarter Exam Non Right Triangle Trigonometry STUDY GUIDE

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3rd Quarter Exam: Non-Right Triangle Trigonometry STUDY GUIDE

AREA FORMULA
If we substitute this new expression for the height, we can write
the triangle area formula as:
(where a and b are adjacent sides and C is the included angle)
We have just discovered that the area of a triangle can be expressed using the lengths of two sides and the
sine of the included angle. This is often referred to as the SAS Formula for the area of a triangle.
The "letters" in the formula may change from problem to problem, so try to remember the pattern of "two
sides and the sine of the included angle".
Example 1:
Given the triangle at the right, find its area.
Express the area rounded to three decimal
places.
Be careful!!! When
using your graphing
calculator, be sure that
you are in DEGREE
Mode, or that you are
using the degree
symbol.
Example 2:
Given the parallelogram shown at the right,
find its EXACT area.
If we are looking for an EXACT answer, we
do NOT want to round our value for sin 60º.
We need to remember that the sin 60º
(from our 30º- 60º- 90º reference triangle)
is
.
Now, the diagonal of a parallelogram divides the
parallelogram into two congruent triangles. So
the total area of the parallelogram will be double
the area of one of the triangles formed by a
diagonal.
square units.

LAW OF SINES
The ratios of each side to the sine of its "partner" are equal to each other.
Law of Sines
Or
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).
Example 1:
In
, side a = 8, m<A = 30º and m<C = 55º. Find side c to the
nearest tenth of an integer.
Since this problem refers to two angles and two
sides, use the Law of Sines.
This answer makes sense, since the larger side is opposite the larger angle.
Example 2:
Find the length of side d.
Again, we are working with two sides and two angles.
Use the Law of Sines:
If the problem asks to find a missing angle, there is another step required for the
solution.
Example 3:
In the diagram, a = 55, c = 20, and m<A = 110º. Find the measure
of <C to the nearest degree.
Using the Law of Sines:
Unfortunately, this is NOT the answer!!
NEW STEP: Using C = sin-1(.342), we have C = 19.999 = 20º

LAW OF COSINES
With the diagram labeled at the left,
the Law of Cosines is as follows:
Notice that <C and side c are at opposite ends of the
formula. Also, notice the resemblance (in the beginning of
the formula) to the Pythagorean Theorem.
We can write the Law of Cosines for each angle around the triangle. Notice in each statement
how the pattern of the letters remains the same.
Law of Cosines
The Law of Cosines can be used to find a missing side for a triangle, or a missing angle.
Example 1:
In
the nearest integer.
, side b = 12, side c = 20 and m<A = 45º. Find side a to
This problem involves all three sides but only one
angle of the triangle. This fits the profile for the
Law of Cosines.
Since the only known angle is A, we use the
version of the Law of Cosines dealing with
angle A.
Example 2:
Find the largest angle, to the nearest tenth of a degree, of a triangle
whose sides are 9, 12 and 18.
In a triangle, the largest angle is opposite the
largest side. We need to find <B.
Use the Law of Cosines:

FORCES
A force is an example of a vector quantity. A vector quantity is a quantity that has both
magnitude (size) and direction. A force is represented by a directed line segment, or vector, in
which the length of the line segment represents the magnitude and an arrow represents the
direction. When two forces act at a point, the single force that has the same effect as the
combination of the applied forces is called the resultant.
applied
force 1
applied
force 2
 The vectors that represent the applied forces form two adjacent sides of a
parallelogram, and the vector that represents the resultant force is the diagonal of
the parallelogram.
Note: The resultant does not bisect the angle between two applied forces that are unequal in
magnitude