Problems using Trigonometry and Pythagoras' Theorem

The sum of the angles of any triangle is 180^{o}.
If you know 2 of the three angles of a right triangle subtract
the sum of the two angles from 180^{o} to calculate
the third angle.

There are 18 basic types of questions you can
solve with the Pythagorean Theorem and
trigonometry. The first 3 types in
the "Questions" section are devoted to Pythagoras's
Theorem. If you know 2 of the three sides
of a right triangle use the Pythagorean Theorem to calculate
the third side.

The last 15 types involve trigonometry functions.
These types are also organized into groups of three similar
functions. After completing two of these you will
probably know what to do in the third one before looking
at the examples.

The "Introduction" reviews
the Pythagorean theorem. "Naming Sides" reviews
the trigonometry triangle names required to use "Soh
Cah Toa". The "Practice" section has 5 sections
of problem types.

Problem Types 1-3: Pythagorean Theorem

Problem Types 4-6: Calculate the sine,
cosine and tangent ratios given one reference angle.

Problem Types 7-9: Calculate the sine,
cosine and tangent ratios given two appropriate sides.

Problem Types 10-12: Calculate the
reference angle given two appropriate sides using trigonometry
functions.

Problem Types 13-18: Calculate the length
of one of the sides given appropriate side and reference
angle using trigonometry functions.

Study these examples:

Example
1

Since ABC in the diagram above is a right triangle, we
can use the Pythagorean Theorem:

a^{2} + b^{2} = c^{2}

12^{2} + b^{2} = 20^{2}

b^{2} = 20^{2} - 12^{2}

b^{2} = 400 - 144 = 256

b = 256 =16 cm

We can use any two sides and
the appropriate trigonometry ratio to find one of the missing
sides.

tan A = opp/adj = a/b = 12/16 = 0.75

<A = tan^{-1}(0.75)

<A = 0.75 =

<A = 36.9^{o} (nearest tenth)

The sum of the angles in a triangle
is 180^{o}.

A + B + C = 180

36.9 + B + 90 = 180

B = 53.1^{o}

Example 2

ABCD is the end wall in an attic. Segment
AB is the sloping ceiling.

AD = 1600 mm; <A = 56 degrees

Ramona decides to hide a large gift box, DEGF, in the attic.
If the box is 600 mm high, what is its length?

Problems
using Trigonometry and Pythagoras' Theorem

The following is an earlier version of the
flash object included in the previous section. It
will be useful to see questions generated where the triangle
is not to scale.