AS Level Biology 9700

5. Trigonometry

Written by: Tharun Athreya
Formatted by: Tharun Athreya

5.1 Angles between 0° and 90°

  • Trigonometry suggests the relationship between side lengths and angles of triangles.
  • The three trigonometric functions Sine, Cosine, and Tangent come from ratios of side lengths in right-angled triangles.
  • \( \sin \theta = \frac{\text{OPPOSITE}}{\text{HYPOTENUSE}} \)
    \( \cos \theta = \frac{\text{ADJACENT}}{\text{HYPOTENUSE}} \)
    \( \tan \theta = \frac{\text{OPPOSITE}}{\text{ADJACENT}} \)
    • Tip: SOHCAHTOA is an easy way to remember the three basic trigonometry ratios.
  \(0^\circ\) \(30^\circ\) \(45^\circ\) \(60^\circ\) \(90^\circ\)
\(\sin(\theta)\) \(0\) \(\frac{1}{2}\) \(\frac{1}{\sqrt{2}}\) \(\frac{\sqrt{3}}{2}\) \(1\)
\(\cos(\theta)\) \(1\) \(\frac{\sqrt{3}}{2}\) \(\frac{1}{\sqrt{2}}\) \(\frac{1}{2}\) \(0\)
\(\tan(\theta)\) \(0\) \(\frac{1}{\sqrt{3}}\) \(1\) \(\sqrt{3}\) undefined

5.2 The general definition of an angle

  • An angle is a measure of the rotation of a line segment about a fixed point. 
  • The CAST Diagram:
    • First quadrant: All graphs are positive between \(0^\circ\) and \(90^\circ\).
    • Second quadrant: Only \(\sin\) is positive between \(90^\circ\) and \(180^\circ\).
    • Third quadrant: Only \(\tan\) is positive between \(180^\circ\) and \(270^\circ\).
    • Fourth quadrant: Only \(\cos\) is positive between \(270^\circ\) and \(360^\circ\).
  • The angle is measured from the positive x-direction.
  • An anticlockwise rotation is taken as positive, and a clockwise rotation is taken as negative.
  • The acute angle made with the x-axis is sometimes called the basic angle or the reference angle.
    Tip: To memorize this diagram, “All Students Trust Cambridge”.

5.3 Graphs of trigonometric functions

  • The graph of \(\sin x\):
  • The graph of \(\cos x\):
  • The sine and cosine functions are called periodic functions because they repeat themselves over and over again.
  • The period of a periodic function is defined as the length of one cycle.
    • The sine and cosine functions repeat every \(360^\circ\).
  • The amplitude of a periodic function is defined as the distance between a maximum (or minimum) point and the principal axis.
    • The functions \(y = \sin x\) and \(y = \cos x\) both have amplitude 1.
  • The symmetry of the curve \(y = \sin x\) shows these important relationships:
    • \(\sin(-x) = -\sin(x)\)
    • \(\sin(180^\circ – x) = \sin(x)\)
    • \(\sin(180^\circ + x) = -\sin(x)\)
    • \(\sin(360^\circ – x) = -\sin(x)\)
    • \(\sin(360^\circ + x) = \sin(x)\)
  • The graph of \(\tan x\):
  • The tangent function repeats its cycle every \(180^\circ\), so its period is \(180^\circ\).
  • There are asymptotes in the tangent function. The branches of the graph get closer and closer to the asymptotes without ever reaching them.
    • Asymptotes are found at \(x = 90^\circ\), \(x = 270^\circ\), and \(x = 450^\circ\).
  • The tangent function does not have an amplitude.

Transformations of trigonometric functions

  • The graph of \( y = a\sin(x) \):
    • The equation for the graph: \( y = 2\sin(x) \)
    • Comparison of the graph \( y = \sin(x) \) and \( y = 2\sin(x) \):
      • The graph of \( y = 2\sin(x) \) is a stretch of the graph of \( y = \sin(x) \).
      • It is a stretch, stretch factor 2, parallel to the y-axis.
      • The amplitude of \( y = 2\sin(x) \) is 2 and the period is 360°.
  • The graph of \( y = \sin(ax) \):
    • The equation for the graph: \( y = \sin(2x) \)
    • Comparison of the graph \( y = \sin(x) \) and \( y = \sin(2x) \):
      • The graph of \( y = \sin(2x) \) is also a stretch of the graph of \( y = \sin(x) \).
      • It is a stretch, stretch factor \( \frac{1}{2} \), parallel to the x-axis.
      • The amplitude of \( y = \sin(2x) \) is 1 and the period is 180°.
  • The graph of \( y = a + \sin(x) \):
    • The equation for the graph: \( y = 1 + \sin(x) \)
    • Comparison of the graph \( y = \sin(x) \) and \( y = 1 + \sin(x) \):
      • The graph of \( y = 1 + \sin(x) \) is a translation of the graph of \( y = \sin(x) \).
      • It is a translation of 1 unit upward.
      • The amplitude of \( y = 1 + \sin(x) \) is 1 and the period is 360°.
  • The graph of \( y = \sin(x + a) \):
    • The equation for the graph: \( y = \sin(x + 90) \)
    • Comparison of the graph \( y = \sin(x) \) and \( y = \sin(x + 90) \):
      • The graph of \( y = \sin(x + 90) \) is a translation of the graph of \( y = \sin(x) \).
      • It is a translation of -90° parallel to the x-axis.
      • The amplitude of \( y = \sin(x + 90) \) is 1 and the period is 360°.

5.4 Inverse trigonometric functions

  • The graph of \( y = \sin^{-1}(x) \):
    • Domain: -1 ≤ x ≤ 1
    • Range: -π/2 ≤ y ≤ π/2
  • The graph of y = \cos^{-1}(x):
    • Domain: -1 ≤ x ≤ 1
    • Range: 0 ≤ y ≤ π
  • The graph of y = \tan^{-1}(x):
    • Domain: x ∈ ℝ
    • Range: -π/2 < y < π/2

5.5 Trigonometric Equations

  • Trigonometric identities are used to simplify trigonometric equations before solving them.
  • The two identities you must know are:
    • \( \tan(x) = \frac{\sin(x)}{\cos(x)} \)
    • \( \sin^2(x) + \cos^2(x) = 1 \)
  • Derivation of the identities:
    • \( \tan(x) = \frac{\text{Opposite}}{\text{Adjacent}} \)
    • \( \frac{\sin(x)}{\cos(x)} = \frac{\left(\frac{\text{Opposite}}{\text{Hypotenuse}}\right)}{\left(\frac{\text{Adjacent}}{\text{Hypotenuse}}\right)} = \frac{\text{Opposite}}{\text{Adjacent}} = \tan(x) \)
    • \( \cos^2(x) + \sin^2(x) = 1 \)

\( x^2 + y^2 = r^2 \)
\( \left( \frac{x}{r} \right)^2 + \left( \frac{y}{r} \right)^2 = 1 \), as \( \cos(\theta) = \frac{x}{r} \) and \( \sin(\theta) = \frac{y}{r} \)

🔥  Approach to solve trigonometric equations: 

  • Sketching a graph 
  • Using trigonometric identities 
  • Using the CAST diagram
  • Factorising quadratic trigonometric equations
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