$\theta$ is an angle in a right-angled triangle.
$\tan \theta=\frac{28}{47}$
What is the value of $\theta$ ?
Give your answer to 1 d.p.
Answer (2)
Find the angle in radians using the inverse tangent function: θ = arctan ( 47 28 ) ≈ 0.5369 radians.
Convert the angle from radians to degrees: θ d e g rees = 0.5369 × π 180 ≈ 30.7841 degrees.
Round the angle to 1 decimal place: θ ≈ 30. 8 ∘ .
The value of θ is: 30. 8 ∘ .
Explanation
Problem Analysis We are given that tan θ = 47 28 and we want to find the value of θ in degrees, rounded to 1 decimal place. Since we know the tangent of the angle, we can use the inverse tangent function (arctan) to find the angle in radians. Then, we convert the angle from radians to degrees.
Find angle in radians First, we find the value of θ in radians using the inverse tangent function: θ = arctan ( 47 28 ) Using a calculator, we find that: θ ≈ 0.5369 radians
Convert to degrees Next, we convert the angle from radians to degrees using the conversion factor π 180 :
θ d e g rees = θ r a d ian s × π 180 θ d e g rees = 0.5369 × π 180 θ d e g rees ≈ 30.7841 degrees
Round to 1 d.p. Finally, we round the value of θ to 1 decimal place: θ ≈ 30. 8 ∘ Therefore, the value of θ is approximately 30.8 degrees.
Final Answer The value of θ is approximately 30. 8 ∘ .
Examples
In surveying, trigonometry is used to calculate angles and distances. For example, if a surveyor measures the slope of a hill and knows the ratio of the vertical rise to the horizontal distance (which is the tangent of the angle of elevation), they can use the arctangent function to determine the angle of elevation. This angle is crucial for creating accurate maps and construction plans.
The angle θ , given that tan θ = 47 28 , is approximately 30. 8 ∘ . We find this by calculating the inverse tangent and converting the result from radians to degrees. Finally, we round to one decimal place as requested.
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