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Math lecture 8 (Introduction to Trigonometry)
 

Math lecture 8 (Introduction to Trigonometry)

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    Math lecture 8 (Introduction to Trigonometry) Math lecture 8 (Introduction to Trigonometry) Document Transcript

    • Introduction to Trigonometry Trigonometry (from Greek trigonon "triangle" + metron "measure") Want to Learn Trigonometry? Here is a quick summary. Trigonometry ... is all about triangles. Right Angled Triangle A right-angled triangle (the right angle is shown by the little box in the corner) has names for each side:    Adjacent is adjacent to the angle "θ", Opposite is opposite the angle, and the longest side is the Hypotenuse.
    • Angles Angles (such as the angle "θ" above) can be in Degrees or Radians. Here are some examples: Angle Degrees Radians 90° π/2 __ Straight Angle 180° π Full Rotation 360° 2π Right Angle "Sine, Cosine and Tangent" The three most common functions in trigonometry are Sine, Cosine and Tangent. We will use them a lot! They are simply one side of a triangle divided by another. For any angle "θ": 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... Sine, Cosine and Tangent are often abbreivated to sin, cos and tan. Unit Circle What you just played with is the Unit Circle. It is a circle with a radius of 1 with its center at 0. Because the radius is 1, it is easy to measure sine, cosine and tangent. Here you can see the sine function being made by the unit circle: You can see the nice graphs made by sine, cosine and tangent. Repeating Pattern Because the angle is rotating around and around the circle the Sine, Cosine and Tangent functions repeat once every full rotation.
    • When we need to calculate the function for an angle larger than a full rotation of 2π (360°) we subtract as many full rotations as needed to bring it back below 2π (360°): 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) Likewise if 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:
    • If you know any 3 of the sides or angles you can find the other 3 (except for the three angles case) See Solving Triangles for more details. 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 / Opposit Trigonometric Identities Right Triangle The Trigonometric Identities are equations that are true for Right Angled Triangles ...... if it is not a Right Angled Triangle refer to our Triangle Identities page. Each side of a right triangle has a name:
    • (Adjacent is adjacent to the angle, and Opposite is opposite ... of course!) Important: We are soon going to be playing with all sorts of functions and it can get quite complex, but remember it all comes back to that simple triangle with:     Angle θ Hypotenuse Adjacent Opposite Sine, Cosine and Tangent The three main functions in trigonometry are Sine, Cosine and Tangent. They are just the length of one side divided by another For a right triangle with an angle θ : Sine Function: sin(θ) = Opposite / Hypotenuse Cosine Function: cos(θ) = Adjacent / Hypotenuse Tangent Function: tan(θ) = Opposite / Adjacent Also, if we divide Sine by Cosine we get:
    • So we can also say: tan(θ) = sin(θ)/cos(θ) That is our first Trigonometric Identity. But Wait ... There is More! We can also divide "the other way around" (such as Adjacent/Opposite instead ofOpposite/Adjacent): Cosecant Function: csc(θ) = Hypotenuse / Opposite Secant Function: sec(θ) = Hypotenuse / Adjacent Cotangent Function: cot(θ) = Adjacent / Opposite Example: if Opposite = 2 and Hypotenuse = 4 then sin(θ) = 2/4, and csc(θ) = 4/2 Because of all that we can say: sin(θ) = 1/csc(θ) cos(θ) = 1/sec(θ) tan(θ) = 1/cot(θ) And the other way around: csc(θ) = 1/sin(θ) sec(θ) = 1/cos(θ) And we also have: cot(θ) = cos(θ)/sin(θ) cot(θ) = 1/tan(θ)
    • More Identitites There are many more identities ... here are some of the more useful ones: Opposite Angle Identities sin (-θ) = - sin (θ) cos (-θ) = cos (θ) tan (-θ) = - tan (θ) Double Angle Identities Half Angle Identities Note that "±" means it may be either one, depending on the value of θ/2
    • Angle Sum and Difference Identities Note that means you can use plus or minus, and the opposite sign. means to use the