Trigonometry Revision Notes BTER Polytechnic Semester 1st

 

1.1 Concept of Angles and Measurement of Angles

• Angle: An angle is formed when two rays originate from a common point called the vertex.
• Types of Angles:
  • Acute: Less than 90°
  • Right: 90°
  • Obtuse: Greater than 90° but less than 180°
  • Reflex: Greater than 180° but less than 360°
  • Full Rotation: 360°
• Units of Measurement:
  • Degrees (°): A full circle is 360°.
  • Radians (rad): A full circle is $2\pi$ radians.
  • Gradians (g or gon): A full circle is 400g.
• Conversions:
  • From Degrees to Radians:
    $\theta (radians) = \theta (degrees) \times \frac{\pi}{180}$
  • From Radians to Degrees:
    $\theta (degrees) = \theta (radians) \times \frac{180}{\pi}$
  • From Degrees to Gradians:
    $\theta (gradians) = \theta (degrees) \times \frac{400}{360}$
  • From Gradians to Degrees:
    $\theta (degrees) = \theta (gradians) \times \frac{360}{400}$

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1.2 T-Ratios of Allied Angles (Without Proof)

• Allied Angles: Two angles are said to be allied if their sum is $90^\circ$ or $\frac{\pi}{2}$ radians. For example, if $\theta$ is an angle, the allied angle is $90^\circ - \theta$.

T-Ratios of Allied Angles:

1. $\sin(90^\circ - \theta) = \cos(\theta)$
2. $\cos(90^\circ - \theta) = \sin(\theta)$
3. $\tan(90^\circ - \theta) = \cot(\theta)$
4. $\cot(90^\circ - \theta) = \tan(\theta)$
5. $\sec(90^\circ - \theta) = \csc(\theta)$
6. $\csc(90^\circ - \theta) = \sec(\theta)$

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1.3 Applications of Sum and Difference Formulae (Without Proof)

• Sum and Difference Formulae:

  1. Sine:

     $$\sin(A + B) = \sin A \cos B + \cos A \sin B$$ $$\sin(A - B) = \sin A \cos B - \cos A \sin B$$

  2. Cosine:

     $$\cos(A + B) = \cos A \cos B - \sin A \sin B$$ $$\cos(A - B) = \cos A \cos B + \sin A \sin B$$

  3. Tangent:

     $$\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$$ $$\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$$

• Applications:

  • Simplifying trigonometric expressions
  • Solving integrals and derivatives in calculus
  • Computing angles in problems involving navigation, physics, and engineering

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1.4 Product Formulae (Transformation of Product to Sum, Difference, and Vice Versa)

• Product to Sum Formulae:

  1. $\sin A \sin B = \frac{1}{2} [\cos(A - B) - \cos(A + B)]$
  2. $\cos A \cos B = \frac{1}{2} [\cos(A - B) + \cos(A + B)]$
  3. $\sin A \cos B = \frac{1}{2} [\sin(A + B) + \sin(A - B)]$

• Sum to Product Formulae:

  1. $\sin A + \sin B = 2 \sin \left(\frac{A + B}{2}\right) \cos \left(\frac{A - B}{2}\right)$
  2. $\cos A + \cos B = 2 \cos \left(\frac{A + B}{2}\right) \cos \left(\frac{A - B}{2}\right)$
  3. $\sin A - \sin B = 2 \cos \left(\frac{A + B}{2}\right) \sin \left(\frac{A - B}{2}\right)$

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1.5 T-Ratios of Multiple Angles (2A, 3A)

• Double Angle Formulae:

  1. $\sin(2A) = 2 \sin A \cos A$
  2. $\cos(2A) = \cos^2 A - \sin^2 A$ or $\cos(2A) = 2 \cos^2 A - 1$ or $\cos(2A) = 1 - 2 \sin^2 A$
  3. $\tan(2A) = \frac{2 \tan A}{1 - \tan^2 A}$

• Triple Angle Formulae:

  1. $\sin(3A) = 3 \sin A - 4 \sin^3 A$
  2. $\cos(3A) = 4 \cos^3 A - 3 \cos A$
  3. $\tan(3A) = \frac{3 \tan A - \tan^3 A}{1 - 3 \tan^2 A}$

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1.6 Graphs of $\sin x$, $\cos x$, and $\tan x$

• Graph of $\sin x$:

  • Periodicity: $2\pi$
  • Range: $[-1, 1]$
  • The sine wave oscillates between -1 and 1, starting at 0 when $x = 0$.

• Graph of $\cos x$:

  • Periodicity: $2\pi$
  • Range: $[-1, 1]$
  • The cosine wave is similar to the sine wave but starts at 1 when $x = 0$.

• Graph of $\tan x$:

  • Periodicity: $\pi$
  • Range: All real numbers ($\mathbb{R}$)
  • The tangent function has vertical asymptotes at $\frac{\pi}{2} + n\pi$ for any integer $n$, and its graph shows periodic rises and falls.

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These concepts are foundational for solving problems in trigonometry, physics, engineering, and various other fields of study.