Question

Given sin⁡θ=1/2 determine the value of sec θ. 0°<θ<90°

Answer 1

Answer:

$sec(\theta)=\frac{2}{\sqrt3}$

Step-by-step explanation:

Given $sin(\theta)=\frac{1}{2}$ , to find $sec(\theta)$, it will be helpful to visualize a right triangle (triangle with a 90 degree angle) associated with that particular θ.  There are a few ways to go about this:

A general solution method

All of the basic trigonometric functions, applied to an angle) are a ratio of two specific sides of any right triangle that holds that angle.

Remember that the Sine of an angle is defined specifically, the ratio of the opposite side (the side across from the angle in the Sine function), and the hypotenuse (the side across from the right angle).  You might remember this through SohCahToa

$sin(\theta)=\frac{opp}{hyp}$

In our case, since $sin(\theta)=\frac{1}{2}$ , so  $\frac{opp}{hyp}=\frac{1}{2}$ .  While there are an infinite number of triangles that have that ratio of those sides, they are all "similar" triangles (corresponding angles congruent, and corresponding sides are proportional, yielding common ratios of sides), and for ease, we can consider simply the triangle where the value of the numerator is the length of the opposite side, and the value of the denominator is equal to the hypotenuse.  So, $opp=1$, and $hyp=2$.

While we haven't actually talked about θ yet, we can still set up the triangle that has these sides so that we can visualize what the triangle looks like. (see image)

This triangle represents the triangle for the unknown θ in the original sine function.  We're tasked with finding the secant of that particular unknown θ.

Working toward Secant

Here, it will be helpful to remember either the reciprocal identities for$sec(\theta)=\frac{1}{cos(\theta)}$, or the definition of the secant function $sec(\theta)=\frac{hyp}{adj}$.

I find that most people remember the reciprocal identities more easily than keeping track of the definitions, so, since secant is related to cosine, it will be important to remember that $cos(\theta)=\frac{adj}{hyp}$.  From there, take the reciprocal of the cosine-value to get the secant-value (which matches the definition of the secant function).

Either way, it comes down to knowing the lengths of the side adjacent to theta, and the hypotenuse.  We already know the length of the hypotenuse, so we just need the length of the adjacent side.

Applying the Pythagorean Theorem

Fortunately, because it is a right triangle, the Pythagorean Theorem applies: $a^{2} +b^{2} =c^{2}$  (where c is the length of the hypotenuse, and a & b are the lengths of the legs)

Substituting the known values for the sides we do know...$(adj)^{2} +(1)^{2} =(2)^{2}\\(adj)^{2} +1 =4$

...isolating "adj" by subtraction...

$(adj)^2=4-1\\(adj)^2=3$

...applying the square root property...

$adj=\sqrt{3}$  or  $adj=-\sqrt{3}$

Identifying which Quadrant the triangle is in

Since we were given that $0^o < \theta < 90^o$, our triangle is an acute triangle (as drawn in the diagram), and is in quadrant I (indicating that both legs will be measured with a positive value.

Thus, we discard the negative solution and conclude that $adj=\sqrt{3}$.

Finding the final solution

From there, $cos(\theta)=\frac{adj}{hyp}$ implies $cos(\theta)=\frac{\sqrt3}{2}$, and through the reciprocal relationship (or simply the definition of secant, whichever is easier for you to remember), $sec(\theta)=\frac{2}{\sqrt3}$

Note:  This method did not require knowing what the angle θ was.

Alternative method using the Unit Circle

If you know well the values of special triangles in the unit circle, you may have identified that $sin(\theta)=\frac{1}{2}$  is associated with $\theta=30^o$.  If so, if you also recall that the ordered pair associated with that point on the unit circle is $(\frac{\sqrt3}{2} ,\frac{1}{2} )$, and that the $cos(\theta)=x\text{-coordinate on the unit circle}$, then you can quickly identify that  $cos(\theta)=\frac{\sqrt3}{2}$.

This method still ends the same: recalling the reciprocal relationship between cosine and secant, giving $sec(\theta)=\frac{2}{\sqrt3}$.