Given that $\cos \beta=-\frac{1}{\sqrt{5}}$, where $180^{\circ}<\beta<360^{\circ}$. Determine, with the aid of a sketch and without using a calculator, the value of $\sin \beta$.
Answer (1)
Determine that β is in the third quadrant because cos β < 0 and 18 0 ∘ < β < 36 0 ∘ .
Use the Pythagorean identity sin 2 β + cos 2 β = 1 and the given value of cos β = − 5 1 to find sin 2 β = 5 4 .
Take the square root to find sin β = ± 5 2 .
Choose the negative value since sin β < 0 in the third quadrant: − 5 2 .
Explanation
Determine the quadrant of beta We are given that cos β = − 5 1 and 18 0 ∘ < β < 36 0 ∘ . We want to find the value of sin β without using a calculator. Since β lies between 18 0 ∘ and 36 0 ∘ , it is in either the third or fourth quadrant. In the third quadrant, both sine and cosine are negative, while in the fourth quadrant, cosine is positive and sine is negative. Since cos β is negative, β must be in the third quadrant, where 18 0 ∘ < β < 27 0 ∘ .
Use the Pythagorean identity We know that sin 2 β + cos 2 β = 1 . We are given cos β = − 5 1 . Substituting this into the identity, we get: sin 2 β + ( − 5 1 ) 2 = 1 sin 2 β + 5 1 = 1 sin 2 β = 1 − 5 1 sin 2 β = 5 4
Solve for sin beta Taking the square root of both sides, we get: sin β = ± 5 4 = ± 5 2 Since β is in the third quadrant, sin β must be negative. Therefore, sin β = − 5 2
Final Answer Thus, the value of sin β is − 5 2 .
Examples
Understanding trigonometric functions like sine and cosine is crucial in fields like navigation. For example, sailors use angles and trigonometric functions to determine their position and direction relative to landmarks or celestial bodies. Knowing the cosine of an angle and the quadrant it lies in allows them to calculate the sine of the angle, which is essential for accurate navigation and course plotting.
