Find sec 𝜃, given 𝜃 is in standard position and its terminal side passes through the point four-fifths, three-fifths.
To answer this question, it’s helpful to recall the unit circle. Remember, this circle has a radius of one, and we can add the following values of 𝜃 to our graph by moving in an anticlockwise direction.
We start at zero, then 𝜋 over two, 𝜋 radians, three 𝜋 over two radians, and finally the full turn takes us back to the start, or two 𝜋 radians. The terminal side is the side that determines the angle. In the case of the unit circle, it’s the radius.
Since we know the terminal side passes through the point four-fifths, three-fifths, we can add 𝜃 to our diagram. The ordered pair four-fifths, three-fifths lies in the first quadrant. So, the value of 𝜃 is between zero and 𝜋 over two radians.
Now this ordered pair actually tells us the dimensions of our right-angled triangle. Since the 𝑥-value is four-fifths, the length of the side adjacent to the angle 𝜃 is four-fifths. The 𝑦-value in our ordered pair is three-fifths, so the length of the side opposite to the angle 𝜃 is three-fifths.
Remember, it’s a unit circle. The radius of this circle is one, so the length of the hypotenuse is also one. This helps us hugely since we can work out the values of sin, cos, and tan 𝜃 using standard right angle trigonometry. But how can we find the value of sec 𝜃?
Well, sec 𝜃 is one over cos 𝜃, so we’ll first find the value of cos 𝜃. Cos 𝜃 is equal to adjacent over hypotenuse. We can label our right-angled triangle relative to the angle 𝜃 to get the opposite is three-fifths, the adjacent is four-fifths, and the hypotenuse is one.
Substituting these values into the formula for cos 𝜃 gives us cos 𝜃 is equal to four-fifths divided by one, which is simply four-fifths. Since sec 𝜃 is one over cos 𝜃, for our value of 𝜃, sec 𝜃 is one over four-fifths, or one divided by four-fifths.
To divide by a fraction, we change the divide sign to a multiply and find the reciprocal of the second fraction. That’s one multiplied by five over four, which is simply five over four. Sec 𝜃 then is five over four.