Chapter Test
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Graph the angle in standard position, find its reference angle, and determine the sign of secant in the quadrant where the terminal side is located.
2
We want to find the exact value of sec - 300^(∘). To do so, let's start by recalling some trigonometric values for special angles.
| Trigonometric Values for Special Angles | |||||
|---|---|---|---|---|---|
| Sine | Cosine | Tangent | Cosecant | Secant | Cotangent |
| sin 30^(∘)=1/2 | cos 30^(∘)=sqrt(3)/2 | tan 30^(∘)=sqrt(3)/3 | csc 30^(∘)=2 | sec 30^(∘)=2sqrt(3)/3 | cot 30^(∘)=sqrt(3) |
| sin 45^(∘)=sqrt(2)/2 | cos 45^(∘)=sqrt(2)/2 | tan 45^(∘)=1 | csc 45^(∘)=sqrt(2) | sec 45^(∘)=sqrt(2) | cot 45^(∘)=1 |
| sin 60^(∘)=sqrt(3)/2 | cos 60^(∘)=1/2 | tan 60^(∘)=sqrt(3) | csc 60^(∘)=2sqrt(3)/3 | sec 60^(∘)=2 | cot 60^(∘)=sqrt(3)/3 |
Next, let's graph θ =- 300^(∘) in standard position so that we can find its reference angle. This way we can use the values from our table. Recall that when an angle has a negative measure, it is being measured counterclockwise.
If the measure of θ is greater than 360^(∘) or less than 0^(∘), we use a coterminal angle with a positive measure between 0^(∘) and 360^(∘) to find the reference angle. To do so, we will add 360^(∘) to the given angle.
We can see that the terminal side of the angle is located in Quadrant I. Therefore, its reference angle θ ' is equal to the coterminal angle which we have just found.
Next, we will recall the signs of the six trigonometric functions in the different quadrants of the coordinate plane.
In Quadrant I — the quadrant where the terminal side of the angle is located — secant is positive. With this information, we can write an equation relating the secant of the angle and the secant of its reference angle. sec - 300^(∘) = sec 60^(∘) Using our table, we can see that sec 60^(∘)= 2. sec - 300 ^(∘) = sec 60^(∘) ⇒ sec - 300 ^(∘) = 2