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":                                      (Sine, Cosine and Tangent are often abbreviated to sin, cos and tan.)   Sine Function: sin(θ) = Opposite / Hypotenuse Cosine Function: cos(θ) = Adjacent / Hypotenuse Tangent Function: tan(θ) = Opposite / Adjacent

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° 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 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 right-angled triangles. The Triangle Identities are equations that are true for all triangles (they don't have to have a right angle).

SWYK TIME!!!