Trigonometry Lesson: Introduction & Basics
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Trigonometry Lesson: Introduction & Basics

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This lesson is the second of the series I am working on. It really should have come first, though. This lesson introduces trigonometry, detailing what it is, what is uses and a few important topics ...

This lesson is the second of the series I am working on. It really should have come first, though. This lesson introduces trigonometry, detailing what it is, what is uses and a few important topics and formulas you'll find yourself using quite frequently.

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Trigonometry Lesson: Introduction & Basics Trigonometry Lesson: Introduction & Basics Presentation Transcript

  • Trigonometry Lesson Two: Introduction to Trigonometry The Basics
  • Trigonometry Is…
    • … the measurement of triangles; it is usually studied as measurements of sides and angles of triangles and as points on a unit circle.
  • Important Things to Remember
    • Pythagorean Theorem
    • Special Right Triangles
    • Trigonometric Functions
    • Law of Cosines & Sines
    • Unit Circle
    • Identities
    • Half & Double Angle Formulas
    • I will go over these topics and some examples in the next several slides.
  • Pythagorean Theorem
    • The Pythagorean Theorem is really easy once you get the hang of it.
    • (leg² + leg² = hypotenuse²) OR (a² + b² = c²)
    • 1. When the two leg lengths are known (a & b), square the length of the each leg, add these two squares together, and square root the resulting sum.
    • 2. When the length of the hypotenuse (c) and either leg is known (a or b), square the length of the hypotenuse, square the length of the leg, subtract these two squares and square root the resulting difference.
    • Example 1 Example 2
  • Special Right Triangles
    • There are two special right triangles: a 30-60-90 and a 45-45-90.
    • 30°-60°-90°
    • These triangles have side lengths with ratios of 1:√3:2, which means the longest leg is always √3 times the length of the shortest leg, and the hypotenuse is always 2 times the shortest leg.
    • 45°-45°-90°
    • These triangles have side lengths with ratios of 1:1:√2, which means that the two legs have the same length (if two angles of a right triangle are equal, then the two legs are equal) and the hypotenuse is √2 times the length of either leg.
  • “ Trig” Functions
    • The “trig” functions of an angle are related to the ratios of the sides of a right angle. They are defined with sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), contangent (cot) and theta ( ϴ ).
    • *opposite leg (opp), hypotenuse (hyp) and adjacent leg (adj).
  • Law of Cosine
    • c² = a² + b² - 2ab cos C
    • Example:
    • x² = 5² + 8² - 2(5)(8) cos(70)
    • x² = 89 – 80 cos(70)
    • x² = √61.64
    • X = 7.85
  • Law of Sine
    • Example:
    • sin40• 10 = x • sin40
    • sin54 sin40
    • sin40•10 = X
    • sin54
    • X ≈ 7.945
  • The Unit Circle
  • Identities
    • Pythagorean Identities Reduction Formulas
    • sin² ϴ + cos² ϴ = 1 sin(- ϴ) = -sin ϴ sin ϴ = -sin( ϴ - )
    • tan² ϴ + 1 = sec² ϴ cos(- ϴ) = cos ϴ cos ϴ = -cos( ϴ - )
    • cot² ϴ + 1 = csc² ϴ tan(- ϴ) = -tan ϴ tan ϴ = -tan( ϴ - )
    • Sum/Difference of Two Angles Reciprocal Identities
    • sin(ϴ ± ø) = sin ϴ cos ø ± cos ϴ sin ø csc ϴ = 1 sec ϴ = 1
    • cos(ø ±ϴ) = cos ϴ cos ø ± sin ϴ sin ø sin ϴ cos ϴ
    • cot ϴ = 1
    • tan(ϴ ± ø) = tan ϴ ± tan ø tan ϴ
    • 1 ± tan ϴ tan ø
    • Quotient Identities
    • tan ϴ = sin ϴ cot ϴ = cos ϴ
    • cos ϴ sin ϴ
  • Half & Double Angle Formulas
    • Half and Double Angle Formulas come in handy when you start to focus on the Identities part of Trig, seeing as they are a branch OF identities.
    • Half-Angle Identities
    • sin t /2 = ± √1- cos t OVER 2
    • cos t /2 = ± √1+ cos t OVER 2
    • tan t /2 = ± √1- cos t OVER 1+ cos t
    • Double-Angle Identities
    • sin2 t = 2sin t cos t
    • cos2 t = cos² t - sin² t
    • cos2 t = 1 – 2sin² t
    • cos2 t = 2cos² t – 1
    • tan2 t = 2tan t OVER 1-tan² t
    • sin²2 t = 1 – cos2 t OVER 2
    • cos² t = 1+ cos2 t OVER 2