Given a function value of an acute angle, find the other five trigonometric function values.
$\csc \theta=1.6$
$\sin \theta \approx \square \cos \theta \approx \square \tan \theta \approx \square$
(Round to two decimal places as needed.)
Answer (1)
Find sin θ using the reciprocal identity: sin θ = c s c θ 1 = 1.6 1 = 0.625 .
Calculate cos θ using the Pythagorean identity: cos θ = 1 − sin 2 θ = 1 − 0.62 5 2 ≈ 0.78 .
Determine tan θ using the quotient identity: tan θ = c o s θ s i n θ = 0.78 0.625 ≈ 0.80 .
Round the values to two decimal places: sin θ ≈ 0.63 , cos θ ≈ 0.78 , tan θ ≈ 0.80 .
Explanation
Problem Analysis We are given that csc θ = 1.6 and that θ is an acute angle. Our goal is to find the values of sin θ , cos θ , and tan θ , rounded to two decimal places.
Calculate sin θ Since csc θ is the reciprocal of sin θ , we have sin θ = csc θ 1 = 1.6 1 = 0.625
Calculate cos θ Now we can find cos θ using the Pythagorean identity sin 2 θ + cos 2 θ = 1 . Since θ is an acute angle, cos θ is positive. Thus, cos θ = 1 − sin 2 θ = 1 − ( 0.625 ) 2 = 1 − 0.390625 = 0.609375 ≈ 0.7806
Calculate tan θ Next, we find tan θ using the identity tan θ = c o s θ s i n θ .
tan θ = 0.7806 0.625 ≈ 0.8006
Round to two decimal places Finally, we round the values to two decimal places: sin θ ≈ 0.63 cos θ ≈ 0.78 tan θ ≈ 0.80
Final Answer Therefore, the values of the trigonometric functions are approximately: sin θ ≈ 0.63 cos θ ≈ 0.78 tan θ ≈ 0.80
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 sine, cosine, and tangent of those angles allows you to calculate the unknown height accurately. Similarly, sailors use trigonometric functions to chart courses and determine distances, ensuring they stay on the correct path and avoid hazards.
