Trigonometry Lecture Notes

Introduction to Trigonometry

• Trigonometry is applied in the context of a right-angled triangle.
• Use the acronym SOHCAHTOA to remember the trigonometric ratios:
  • Sine (SOH): Opposite/Hypotenuse
  • Cosine (CAH): Adjacent/Hypotenuse
  • Tangent (TOA): Opposite/Adjacent

Solving for Sides in Right-Angled Triangles

Example 1: Using Sine to Find Hypotenuse

1. Label sides: Opposite, Hypotenuse, Adjacent.
2. Identify which sides are involved (ignore the unnecessary side).
3. Use Sine (SOH) since you have Opposite and want Hypotenuse.
4. Formula: Hypotenuse = Opposite / Sine(angle)
5. Calculate:
   • Opposite = 6, angle = 43°
   • Hypotenuse = 6 / sin(43)
   • Result: 8.79 cm (rounded to 8.8 cm)

Example 2: Using Sine to Find Opposite

1. Label sides and identify necessary ones.
2. Use Sine (SOH) with Opposite and Hypotenuse.
3. Formula: Opposite = Hypotenuse * Sine(angle)
4. Calculate:
   • Hypotenuse = 9, angle = 36°
   • Opposite = 9 * sin(36)
   • Result: 5.29 cm (rounded to 5.3 cm)

Solving for Angles in Right-Angled Triangles

Example: Using Cosine to Find Angle

1. Identify sides: Hypotenuse and Adjacent.
2. Use Cosine (CAH) since O is not needed.
3. Formula: cos(angle) = Adjacent/Hypotenuse
4. Calculate:
   • Adjacent = 7, Hypotenuse = 13
   • cos(angle) = 7/13
   • Use inverse cosine to find the angle.
   • Result: 57.4°

Special Non-Calculator Values

• sin(30) = 1/2
• cos(60) = 1/2

Non-Right Angled Triangles

Sine Rule

• Applicable when there are two pairs of opposite sides and angles.
• Formula: ( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} )

Example: Finding a Side

• Label triangle, use known sides/angles.
• Solve for unknown using proportion form.

Example: Finding an Angle

• Use inverse sine after setting up proportion with known sides/angles.

Cosine Rule

• Use when opposite pairs are not available.
• Formula for sides: ( a^2 = b^2 + c^2 - 2bc \cdot \cos(A) )
• Formula for angles: Rearrange cosine rule to solve for angle.

Area of a Triangle (Non-Right Angled)

• Formula: ( \text{Area} = \frac{1}{2}ab \cdot \sin(C) )
• Use known side lengths and included angle.

Example: Given Area, Find Angle

• Rearrange area formula to solve for angle.
• Use inverse sine to find angle value.

Conclusion

• Understand when to apply SOHCAHTOA, sine rule, and cosine rule.
• Practice with different triangle configurations to strengthen skills.

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