Given a function value of an acute angle, find the other five trigonometric function values.
$\csc \theta=1.7$
$\sin \theta \approx 0.59 \quad \cos \theta \approx 0.81 \quad \tan \theta \approx 0.73$
(Round to two decimal places as needed.)
$\sec \theta \approx \square \cot \theta \approx \square$
(Round to two decimal places as needed.)
Answer (1)
Use the reciprocal identity sec θ = c o s θ 1 and the given value cos θ ≈ 0.81 to calculate sec θ .
Use the reciprocal identity cot θ = t a n θ 1 and the given value tan θ ≈ 0.73 to calculate cot θ .
Calculate sec θ ≈ 0.81 1 ≈ 1.2345679 and round to two decimal places: sec θ ≈ 1.23 .
Calculate cot θ ≈ 0.73 1 ≈ 1.369863 and round to two decimal places: cot θ ≈ 1.37 . The final answers are sec θ ≈ 1.23 and cot θ ≈ 1.37 .
Explanation
Analyze the given information We are given the following trigonometric values for an acute angle θ :
csc θ = 1.7 sin θ ≈ 0.59 cos θ ≈ 0.81 tan θ ≈ 0.73
We need to find the values of sec θ and cot θ , rounded to two decimal places.
Calculate sec(theta) To find sec θ , we use the reciprocal identity: sec θ = cos θ 1 Since cos θ ≈ 0.81 , we have: sec θ ≈ 0.81 1 ≈ 1.2345679 Rounding to two decimal places, we get: sec θ ≈ 1.23
Calculate cot(theta) To find cot θ , we use the reciprocal identity: cot θ = tan θ 1 Since tan θ ≈ 0.73 , we have: cot θ ≈ 0.73 1 ≈ 1.369863 Rounding to two decimal places, we get: cot θ ≈ 1.37
State the final answer Therefore, the values of sec θ and cot θ , rounded to two decimal places, are: sec θ ≈ 1.23 cot θ ≈ 1.37
Examples
Trigonometric functions are incredibly useful in fields like navigation and surveying. For example, if you're a surveyor trying to determine the height of a building using angles and distances, knowing the values of trigonometric functions like secant and cotangent for a given angle allows you to calculate these heights accurately. These calculations are essential for creating accurate maps and ensuring structural safety.
