Find [tex]$\sin \theta$[/tex]. [tex]$\cot \theta=-\frac{2}{5}, \theta$[/tex] in quadrant IV. [tex]$\sin \theta=$[/tex]
Answer (1)
We are given cot θ = − 5 2 and θ is in quadrant IV.
We find tan θ = − 2 5 .
Using the identity 1 + tan 2 θ = sec 2 θ , we find sec θ = ± 2 29 . Since θ is in quadrant IV, 0"> cos θ > 0 , so cos θ = 29 2 .
Using the identity sin 2 θ + cos 2 θ = 1 , we find sin θ = − 29 5 29 .
− 29 5 29
Explanation
Analyze the given information We are given that cot θ = − 5 2 and θ is in quadrant IV. We want to find sin θ . Since cot θ = s i n θ c o s θ = − 5 2 , we know that tan θ = c o s θ s i n θ = − 2 5 .
Use the identity 1 + tan 2 θ = sec 2 θ We can use the trigonometric identity 1 + tan 2 θ = sec 2 θ . Since tan θ = − 2 5 , we have tan 2 θ = ( − 2 5 ) 2 = 4 25 . Therefore, sec 2 θ = 1 + 4 25 = 4 4 + 4 25 = 4 29 .
Determine cos θ Taking the square root of both sides, we get sec θ = ± 2 29 . Since sec θ = c o s θ 1 , we have cos θ = ± 29 2 . Because θ is in quadrant IV, cos θ is positive, so cos θ = 29 2 .
Use the identity sin 2 θ + cos 2 θ = 1 Now we use the identity sin 2 θ + cos 2 θ = 1 to find sin θ . We have sin 2 θ = 1 − cos 2 θ = 1 − ( 29 2 ) 2 = 1 − 29 4 = 29 29 − 29 4 = 29 25 .
Determine sin θ Taking the square root of both sides, we get sin θ = ± 29 5 . Since θ is in quadrant IV, sin θ is negative, so sin θ = − 29 5 . To rationalize the denominator, we multiply the numerator and denominator by 29 to get sin θ = − 29 5 29 .
Final Answer Therefore, sin θ = − 29 5 29 .
Examples
Understanding trigonometric functions like sine and cotangent is crucial in fields like navigation. For example, sailors use angles and trigonometric relationships to determine their position and direction relative to landmarks or celestial bodies. Knowing the cotangent of an angle can help calculate the sine, which is essential for determining distances and bearings.
