Trigonometry II

Grade 12 Trigonometry Lesson

Lesson Objectives

  • Understand and apply the sine, cosine, and tangent ratios.
  • Solve for unknown sides and angles in right-angled triangles.

Lesson Introduction

In right-angled triangles, the basic trigonometric ratios are:

  • Sine: \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
  • Cosine: \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
  • Tangent: \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}

Worked Examples

Sine Ratio Examples

Example 1: Given \theta = 30^\circ , \text{hypotenuse} = 10 , find the opposite side.
\text{opposite} = 10 \cdot \sin(30^\circ) = 5
Example 2: \sin(\theta) = \frac{4}{5}, \text{hypotenuse} = 10 . Find \theta .
\theta = \sin^{-1}\left(\frac{4}{5}\right) \approx 53.13^\circ
Example 3: \theta = 45^\circ, \text{opposite} = 7 . Find hypotenuse.
\text{hypotenuse} = \frac{7}{\sin(45^\circ)} \approx 9.9
Example 4: Opposite = 6, hypotenuse = 10. Find angle A.
A = \sin^{-1}\left(\frac{6}{10}\right) \approx 36.87^\circ
Example 5: \theta = 60^\circ, \text{hypotenuse} = 12 . Find opposite.
\text{opposite} = 12 \cdot \sin(60^\circ) \approx 10.39

Cosine Ratio Examples

Example 1: \theta = 60^\circ, \text{hypotenuse} = 10 . Find adjacent.
\text{adjacent} = 10 \cdot \cos(60^\circ) = 5
Example 2: \cos(\theta) = \frac{3}{5} . Find \theta .
\theta = \cos^{-1}\left(\frac{3}{5}\right) \approx 53.13^\circ
Example 3: Adjacent = 4, \theta = 30^\circ . Find hypotenuse.
\text{hypotenuse} = \frac{4}{\cos(30^\circ)} \approx 4.62
Example 4: \cos(\theta) = 0.8 . Find \theta .
\theta = \cos^{-1}(0.8) \approx 36.87^\circ
Example 5: \theta = 45^\circ, \text{hypotenuse} = 8 . Find adjacent.
\text{adjacent} = 8 \cdot \cos(45^\circ) \approx 5.66

Tangent Ratio Examples

Example 1: \tan(45^\circ) = 1 . If adjacent = 6, opposite = 6.
Example 2: \tan(\theta) = \frac{3}{4} . Find \theta .
\theta = \tan^{-1}\left(\frac{3}{4}\right) \approx 36.87^\circ
Example 3: \theta = 60^\circ, \text{adjacent} = 5 . Find opposite.
\text{opposite} = 5 \cdot \tan(60^\circ) \approx 8.66
Example 4: Opposite = 9, adjacent = 7. Find \theta .
\theta = \tan^{-1}\left(\frac{9}{7}\right) \approx 52.43^\circ
Example 5: \theta = 30^\circ, \text{adjacent} = 10 . Find opposite.
\text{opposite} = 10 \cdot \tan(30^\circ) \approx 5.77

Exercises

  1. [WAEC] Given \theta = 40^\circ and hypotenuse = 15, find the opposite side. (Past Question)
  2. If \cos(\theta) = \frac{5}{13} , find \theta .
  3. Solve: \tan(\theta) = \frac{7}{24} . Find \theta .
  4. Given opposite = 10, hypotenuse = 20. Find \theta .
  5. Find the adjacent side if \theta = 60^\circ and hypotenuse = 14.
  6. [WAEC] If \sin(\theta) = 0.6 , find \theta . (Past Question)
  7. [NECO] Adjacent = 8, \theta = 45^\circ . Find hypotenuse. (Past Question)
  8. Given opposite = 9 and adjacent = 12, find \theta .
  9. Solve: \cos(30^\circ) = \frac{\text{adjacent}}{x} , if x = 10 .
  10. [JAMB] Find the opposite side if \theta = 75^\circ and adjacent = 6. (Past Question)

Conclusion/Recap

By mastering sine, cosine, and tangent ratios, you can solve real-world and mathematical problems involving right-angled triangles. Always use inverse functions to find angles, and direct ratios to find missing sides.

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