Trigonometry

TRIGONOMETRY

Aileen Grace Delima

TRIGONOMETRY

• is among the earliest applications of Euclidean geometry.

• branch of mathematics that deals with the relationships between the sides and angles of triangles and with the properties and applications of the trigonometric functions of angles.

TRIGONOMETRY

• The two branches of trigonometry are:

• Plane trigonometry - which deals with figures lying wholly in a single plane

• Spherical trigonometry - which deals with triangles that are sections of the surface of a sphere

TRIGONOMETRY

• deal with problems in astronomy and has many other uses to the measurement of distance that are difficult or impossible to measure directly.

• consider problems involving periodic phenomena such as sound waves.

Parts of a Triangle

• Hypotenuse• The side opposite to

the right angle in a triangle is called the hypotenuse.

• Here the side AC is the hypotenuse.

Parts of a Triangle

• Opposite Side• The side opposite to

the angle in consideration is called the opposite side.

• So, if we are considering angle A, then the opposite side is CB.

Parts of a Triangle

• Base/ Adjacent Side

• The third side of the triangle, which is one of the arms of the angle under consideration, is called the base.

• If A is the angle under consideration, then the side AB is the base.

A

B

C

Similar Triangles

C1 C2 C3

B1

B2

B3

The consequent proportions of corresponding sides can be rewritten as follows:

BC = B1C1 = B2C2 = B3C3 = s(1)

AB AB1 AB3 AB3

AC = A1C1 = A2C2 = A3C3 = c(2)

AB AB1 AB2 AB3

BC = B1C1 = B2C2 = B3C3 = t (3)

AC AC1 AC2 AC3

Now, since angle A is common to all triangles, we can describe the ratios s, c, t with reference to the acute angle A:

s= opposite leg (to A) (1)

hypotenuse

c= adjacent leg (to A) (2)

hypotenuse

t= opposite leg (to A) (3)

adjacent leg (to A)

This means that for the same acute angle A on any right triangle the ratios (1), (2), (3) are constant, hence we can give them names:

Define:sine A = opposite leg sin A

hypotenusecosine A = adjacent leg cos A

hypotenuse tangent A = opposite leg tan A

adjacent leg

These are the three basic trigonometric functions of angle A.

Complementary Angle Property

Observe that angle A and angle B are complementary, i.e., A = B = 90 ° and that

• sin A = cos B

• cos A = sin BWe state the property above as follows:

“If two angles are complementary, then the sine of one equals the cosine of the other.”

Problem: Find the values of sin A, cos A, and tan A.

Solution:

sin(A) = opposite / hypotenuse

= 4.00 cm / 7.21 cm

= 0.5548

Or simply:

sin (A) = 0.5548

Solution:

cos(A) = adjacent / hypotenuse

= 6.00 cm / 7.21 cm

= 0.8322

Or simply:

cos (A) = 0.8322

Solution:

tan(A) = opposite / adjacent

= 4.00 cm / 6.00 cm

= 0.6667

Or simply:

tan (A) = 0.6667

Here is an easy way to remember these relationships for trig functions and the right triangle.

SOH - CAH - TOA

It is pronounced "so - ka - toe - ah".

The SOH stands for "Sine of an angle is Opposite over Hypotenuse."

The CAH stands for "Cosine of an angle is Adjacent over Hypotenuse."

The TOA stands for "Tangent of an angle is Opposite over Adjacent."

Law of Sines

sin A = sin B = sin C

a b c

Or

a = b = c

sin A sin B sin C

Law of Sines

The area of the triangle equals one-half of the product of the lengths of two sides and

the sine of their included angle.

Example

In a triangle ABC, a = 7; b=4; A=75°

Find B.

A = 75°

BC a = 7

b = 4

c

Solution:

a = b

sin A sin B

7/ sin 75° = 4/sin B

4 sin 75 ° = 7 sin B

4(0.9659) = 7 sin B

sin B = 4(0.9659)/ 7

sin B = 0.5519

B = 34°

Law of Cosines

Given a triangle, the square of any length of any side equals the sum of the squares of

the lengths of the other two sides decreased by twice the product of these

two sides and the cosine of their included angle.

Law of Cosines

• In symbols, a2 = b2 + c2 – 2bc cos A

b2 = a2 + c2 – 2ac cos B

c2 = a2 + b2 - 2ab cos C

Example

• Given: In a triangle ABC, a=3; b=5; C=60°

C=60°

BA a = 3

b = 5

c=?

Solution:

c2 = a2 + b2 - 2ab cos 60°

c2 = 32 + 52- 2 (3)(5)(0.5)

= 9 + 25 -15

= 34 – 15

= 19

c = 4.36

TRIVIA

Fourier Ears Only!

• Writing a function as a sum of sines and cosines is called a Fourier series.

• In fact, your ears do Fourier series automatically! • There are little hairs (cilia) in you ears which

vibrate at specific (and different) frequencies. • When a wave enters your ear, the cilia will

vibrate if the wave function "contains" any component of the corresponding frequency!

• Because of this, you can distinguish sounds of various pitches!

Thank you!