Given a function value of an acute angle, find the other five trigonometric function values. [tex]\csc \theta=1.6[/tex] [tex]\sin \theta \approx .63 \cos \theta \approx .78 \tan \theta \approx .80[/tex] (Round to two decimal places as needed.) [tex]\sec \theta \approx \square \cot \theta \approx \square[/tex] (Round to two decimal places as needed.)
Answer (2)
Use the reciprocal identity to find sec θ from cos θ : sec θ = c o s θ 1 .
Substitute the given value of cos θ ≈ 0.78 to find sec θ ≈ 0.78 1 ≈ 1.28 .
Use the reciprocal identity to find cot θ from tan θ : cot θ = t a n θ 1 .
Substitute the given value of tan θ ≈ 0.80 to find cot θ = 0.80 1 = 1.25 . The final answers are sec θ ≈ 1.28 and cot θ = 1.25 .
Explanation
Problem Analysis and Given Data We are given that csc θ = 1.6 , sin θ ≈ 0.63 , cos θ ≈ 0.78 , and tan θ ≈ 0.80 . We need to find the values of sec θ and cot θ , rounded to two decimal places.
Calculating sec θ To find sec θ , we use the reciprocal identity sec θ = c o s θ 1 . Since we are given that cos θ ≈ 0.78 , we have sec θ ≈ 0.78 1 ≈ 1.28205 Rounding to two decimal places, we get sec θ ≈ 1.28 .
Calculating cot θ To find cot θ , we use the reciprocal identity cot θ = t a n θ 1 . Since we are given that tan θ ≈ 0.80 , we have cot θ ≈ 0.80 1 = 1.25 So, cot θ = 1.25 .
Final Answer Therefore, the values of sec θ and cot θ are approximately 1.28 and 1.25, respectively.
Examples
Trigonometric functions are extremely useful in fields like navigation, engineering, and physics. For example, when designing a bridge, engineers use trigonometric functions to calculate angles and forces. Also, in navigation, these functions are used to determine the direction and distance of travel. Understanding these relationships allows for precise calculations and safe designs.
To find sec θ and cot θ from the given angles, we calculated sec θ ≈ 1.28 and cot θ = 1.25 using the reciprocal identities. These values depend on the approximated values of cos θ and tan θ .
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