Download Trigonometry

Trigonometry
By：Holly and Elaine
What is Trigonometry？
Trigonometry is the branch of
mathematics dealing with the relations of
the sides and angles of triangles and with
the relevant functions of any angles. And
Trigonometry (from Greek trigōnon,
"triangle" and metron, "measure") is a
branch of mathematics that studies
relationships involving lengths and angles
of triangles.
Sine,Cosine and Tangent
Trigonometry is good at find a missing side or angle in a triangle.
The special functions Sine, Cosine and Tangent help us!
They are simply one side of a right-angled triangle divided by another.
For any angle "θ":
sin(θ) =
Sine Function: Opposite / Hypotenu
se
cos(θ) =
Cosine Function: Adjacent / Hypoten
use
Tangent Function:
tan(θ) =
Opposite / Adjacent
(Sine, Cosine and Tangent are often abbreviated to sin, cos and tan.)
Example:What is the sine of
35°?
Using this triangle (lengths
are only to one decimal
place):
sin(35°) = Opposite /
Hypotenuse = 2.8/4.9 = 0.57
Right Angled Triangle
The triangle of most interest is the right-angled triangle.
The right angle is shown by the little box in the corner.
We usually know another angle θ.
And we give names to each side:
•Adjacent is adjacent (next to) to the angle θ
•Opposite is opposite the angle θ
•the longest side is the Hypotenuse
Angles can be in Degrees or Radians.
Here are some examples:
Angle
Degrees
Radians
Right Angle
90°
π/2
__Straight Angle
180°
π
Full Rotation
360°
2π
Repeating Pattern
Because the angle is rotating around and
around the circle the Sine, Cosine and
Tangent functions repeat once every full
rotation (see Amplitude, Period, Phase Shift
and Frequency).
When we want to calculate the function for
an angle larger than a full rotation of 360°
(2π radians) we subtract as many full
rotations as needed to bring it back below
360° (2π radians):
Example:
what is the cosine of 370°?
370° is greater than 360° so let us subtract
360°
370° − 360° = 10°
cos(370°) = cos(10°) = 0.985 (to 3 decimal
places)
And when the angle is less than zero, just
add full rotations.
Example: what is the sine of −3 radians?
−3 is less than 0 so let us add 2π radians
−3 + 2π = −3 + 6.283... = 3.283... radians
sin(−3) = sin(3.283...) = −0.141 (to 3
decimal places)
Solving Triangles
A big part of Trigonometry is Solving Triangles.
"Solving" means finding missing sides and angles.
Example: Find the Missing Angle "C"
Angle C can be found using angles of a triangle add
to 180°:
So C = 180° − 76° − 34° = 70°
It is also possible to find missing side lengths and
more. The general rule is:
When we know any 3 of the sides or angles we
can find the other 3
(except for the three angles case)
Other Functions (Cotangent, Secant, Cosecant)
Similar to Sine, Cosine and Tangent, there are
three other trigonometric functions which
are made by dividing one side by another:
Cosecant Function:
csc(θ) =
Hypotenuse / Opposite
Secant Function:
sec(θ) =
Hypotenuse / Adjacent
Cotangent Function: cot(θ) = Adjacent / Opposite
The Trigonometric Identities are
equations that are true for all rightangled triangles.
The Triangle Identities are equations that are true
for all triangles (they don't have to have a right
angle).
SWYK TIME!!!