Given that [tex]$\tan \theta=-1$[/tex], what is the value of [tex]$\sec \theta$[/tex], for [tex]$\frac{3 \pi}{2}\ \textless \ \theta\ \textless \ 2 \pi$[/tex] ?
A. [tex]$-\sqrt{2}$[/tex]
B. [tex]$\sqrt{2}$[/tex]
C. 0
D. 1
Answer (2)
We know that tan θ = − 1 and θ is in the fourth quadrant.
Since tan θ = c o s θ s i n θ = − 1 , we have sin θ = − cos θ .
Using the identity sin 2 θ + cos 2 θ = 1 , we find cos θ = 2 2 .
Therefore, sec θ = c o s θ 1 = 2 .
2
Explanation
Analyze the problem We are given that tan θ = − 1 and 2 3 π < θ < 2 π . This means that θ lies in the fourth quadrant. In the fourth quadrant, cosine is positive and sine is negative. We want to find the value of sec θ .
Use trigonometric identities Since tan θ = − 1 , we have c o s θ s i n θ = − 1 , which implies sin θ = − cos θ . We also know the trigonometric identity sin 2 θ + cos 2 θ = 1 . Substituting sin θ = − cos θ into this identity, we get ( − cos θ ) 2 + cos 2 θ = 1 cos 2 θ + cos 2 θ = 1 2 cos 2 θ = 1 cos 2 θ = 2 1
Determine the sign of cosine Taking the square root of both sides, we get cos θ = ± 2 1 = ± 2 2 Since θ is in the fourth quadrant, cos θ is positive. Therefore, cos θ = 2 2
Calculate secant Now we can find sec θ , which is the reciprocal of cos θ : sec θ = cos θ 1 = 2 2 1 = 2 2 = 2 2 2 = 2
Examples
Understanding trigonometric functions like tangent and secant is crucial in fields like navigation and physics. For instance, when calculating the trajectory of a projectile, the angle of launch and its tangent help determine the range, while the secant can be used to find related distances or forces. These functions are also essential in electrical engineering for analyzing alternating current circuits.
Given that tan θ = − 1 and that θ is in the fourth quadrant, we find that sec θ = 2 . Thus, the correct choice is B. 2 .
;
