Trigonometry

• What is the definition of trigonometry?


  Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles.
• Why is trigonometry particularly useful?

  It is particularly useful when dealing with right-angled triangles.
• How is trigonometry used in real-world applications?

  It is used to measure the height of buildings or calculate distances in navigation.
• What are the three main trigonometric ratios?

  - Sine (sin): opposite / hypotenuse - Cosine (cos): adjacent / hypotenuse - Tangent (tan): opposite / adjacent
• What does the sine ratio represent in a right-angled triangle?

  The sine ratio represents the relationship between the opposite side and the hypotenuse.
• What does the cosine ratio represent in a right-angled triangle?

  The cosine ratio represents the relationship between the adjacent side and the hypotenuse.
• What does the tangent ratio represent in a right-angled triangle?

  The tangent ratio represents the relationship between the opposite side and the adjacent side.
• If you know the angle and the adjacent side, which trigonometric ratio would you use to find the hypotenuse?

  You would use the cosine ratio.
• If you know the opposite side and the hypotenuse, which trigonometric ratio would you use to find the angle?

  You would use the inverse sine function.
• What is the process for using trigonometric ratios in right-angled triangles?

  1. Identify the given information: angle and at least one side. 2. Label the sides: opposite, adjacent, hypotenuse. 3. Choose the appropriate ratio (sin, cos, or tan). 4. Set up the equation using the chosen ratio. 5. Solve the equation to find the unknown value.
• How do you rearrange the sine ratio to solve for the angle?

  You rearrange it to: $\theta =$$\sin^{-1}\left(\frac{\text{opposite}}{\text{hypotenuse}}\right)$.
• How would you find the height of a tree if you know the length of its shadow and the angle of elevation?

  You would use the tangent ratio: $\tan(\theta) =$$\frac{\text{height}}{\text{shadow length}}$.
• How would you find the length of the hypotenuse if you know the opposite side and the angle?

  You would use the sine ratio: $\sin(\theta) =$$\frac{\text{opposite}}{\text{hypotenuse}}$.
• How do you find the height of a kite flying at an angle with the ground?

  You would use the sine ratio: $\sin(\theta) =$$\frac{\text{height}}{\text{hypotenuse}}$.
• What is the process to find the angle a ladder makes with the ground?

  You would use the inverse sine function: $\theta =$$\sin^{-1}\left(\frac{\text{opposite}}{\text{hypotenuse}}\right)$.
• What is the formula for the sine ratio?

  The formula for the sine ratio is $\sin(\theta) =$$\frac{\text{opposite}}{\text{hypotenuse}}$.
• What is the formula for the cosine ratio?

  The formula for the cosine ratio is $\cos(\theta) =$$\frac{\text{adjacent}}{\text{hypotenuse}}$.
• What is the formula for the tangent ratio?

  The formula for the tangent ratio is $\tan(\theta) =$$\frac{\text{opposite}}{\text{adjacent}}$.
• What is the height reached by the ladder against the wall?

  4 metres
• How do you find the angle that the ladder makes with the ground?

  By using the inverse sine (arcsin) function
• If the ladder length is 5 metres and the height on the wall is 4 metres, what is the angle made with the ground?

  Approximately 53°
• What is the relationship between the opposite side and the hypotenuse in this scenario?

  The opposite side is the height on the wall, and the hypotenuse is the ladder length
• What is the formula used to calculate the angle in this scenario?


  $\theta =$$\arcsin\left(\frac{4}{5}\right)$
• Why do we round the angle to the nearest degree?

  To provide a simpler and more practical answer
• What is the angle of depression from the top of a 30-metre tall building to a car on the ground?

  20°
• What does the angle of depression represent in this scenario?


  The angle between the horizontal line from the observer and the line of sight to the car
• How do you find the distance from the base of the building to the car using the angle of depression?

  By using the tangent ratio