Work out the missing side x. Give your answers to 1 decimal place or better. Diagram 1: A right triangle with hypotenuse 4 cm, an angle of 28 degrees, and side x opposite the right angle. x = cm Diagram 2: A right triangle with base 5 cm, an angle of 15 degrees, and side x opposite the right angle. x = cm Diagram 3: A right triangle with hypotenuse 8 cm, an angle of 74 degrees, and side x adjacent to the 74-degree angle. x = cm Diagram 4: A right triangle with hypotenuse x, an angle of 29 degrees, and side 5.4 cm adjacent to the 29-degree angle. x = cm Diagram 5: A right triangle with hypotenuse x, an angle of 42 degrees, and side 3.4 cm opposite the 42-degree angle. x = cm Diagram 6: A right triangle with hypotenuse x, an angle of 46 degrees, and side 5.8 cm adjacent to the 46-degree angle. x = cm
Answer (1)
To solve each of these problems, we use trigonometric ratios, which are applicable in right triangles. Here's how you can find the missing side 'x' in each diagram:
Diagram 1:
Given: Hypotenuse = 4 cm, Angle = 28 degrees, Side opposite angle (x).
Use sine function: sin ( θ ) = hypotenuse opposite
x = sin ( 2 8 ∘ ) × 4
Calculate x ≈ sin ( 2 8 ∘ ) × 4 ≈ 1.879 cm (to 3 decimal places).
Diagram 2:
Given: Base = 5 cm, Angle = 15 degrees, Side opposite angle (x).
Use tangent function: tan ( θ ) = adjacent opposite
x = tan ( 1 5 ∘ ) × 5
Calculate x ≈ tan ( 1 5 ∘ ) × 5 ≈ 1.339 cm (to 3 decimal places).
Diagram 3:
Given: Hypotenuse = 8 cm, Angle = 74 degrees, Side adjacent to angle (x).
Use cosine function: cos ( θ ) = hypotenuse adjacent
x = cos ( 7 4 ∘ ) × 8
Calculate x ≈ cos ( 7 4 ∘ ) × 8 ≈ 2.209 cm (to 3 decimal places).
Diagram 4:
Given: Side adjacent = 5.4 cm, Angle = 29 degrees, Hypotenuse (x).
Use cosine function: cos ( θ ) = hypotenuse adjacent
x = c o s ( 2 9 ∘ ) 5.4
Calculate x ≈ c o s ( 2 9 ∘ ) 5.4 ≈ 6.173 cm (to 3 decimal places).
Diagram 5:
Given: Opposite side = 3.4 cm, Angle = 42 degrees, Hypotenuse (x).
Use sine function: sin ( θ ) = hypotenuse opposite
x = s i n ( 4 2 ∘ ) 3.4
Calculate x ≈ s i n ( 4 2 ∘ ) 3.4 ≈ 5.104 cm (to 3 decimal places).
Diagram 6:
Given: Side adjacent = 5.8 cm, Angle = 46 degrees, Hypotenuse (x).
Use cosine function: cos ( θ ) = hypotenuse adjacent
x = c o s ( 4 6 ∘ ) 5.8
Calculate x ≈ c o s ( 4 6 ∘ ) 5.8 ≈ 8.043 cm (to 3 decimal places).
Each calculation involves a trigonometric function and can be done using a calculator capable of trigonometric functions. Make sure your calculator is set to the degree mode when performing these calculations.
