Given a function value of an acute angle, find the other five trigonometric function values.
$\csc \theta=2.6$
$\sin \theta \approx 0.38 \quad \cos \theta \approx 0.92 \quad \tan \theta \approx 0.42$
(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 to find sec θ from cos θ : sec θ = c o s θ 1 ≈ 0.92 1 ≈ 1.09 .
Use the reciprocal identity to find cot θ from tan θ : cot θ = t a n θ 1 ≈ 0.42 1 ≈ 2.38 .
Round both results to two decimal places.
State the final answers: sec θ ≈ 1.09 and cot θ ≈ 2.38 .
Explanation
Analyze the problem and given data We are given the following trigonometric values for an acute angle θ :
csc θ = 2.6 sin θ ≈ 0.38 cos θ ≈ 0.92 tan θ ≈ 0.42
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 We are given that cos θ ≈ 0.92 , so sec θ ≈ 0.92 1 ≈ 1.0869565 Rounding to two decimal places, we get sec θ ≈ 1.09 .
Calculate cot(theta) To find cot θ , we use the reciprocal identity: cot θ = tan θ 1 We are given that tan θ ≈ 0.42 , so cot θ ≈ 0.42 1 ≈ 2.3809523 Rounding to two decimal places, we get cot θ ≈ 2.38 .
State the final answer Therefore, the approximate values of sec θ and cot θ are: sec θ ≈ 1.09 cot θ ≈ 2.38
Examples
Trigonometric functions are essential in various fields, including physics, engineering, and navigation. For instance, when designing a bridge, engineers use trigonometric functions to calculate angles and distances, ensuring the structure's stability. Similarly, in navigation, sailors and pilots use trigonometric functions to determine their position and direction based on angles to known landmarks or celestial bodies. These calculations rely on the relationships between angles and sides of triangles, making trigonometric functions a fundamental tool in solving real-world problems.
