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Math  /  Trigonometry

QuestionL steps for full marks as required. Correct answer with NO work shown = NO ma
1. Determine the exact values of the six trigonometric ratios for 585∘585^{\circ}. [8.1] [2 marks]

Studdy Solution

STEP 1

1. We need to find the six trigonometric ratios: sine, cosine, tangent, cosecant, secant, and cotangent.
2. The angle given is 585∘ 585^\circ .
3. Trigonometric functions are periodic, and we can use this property to simplify calculations.

STEP 2

1. Reduce the angle 585∘ 585^\circ to an equivalent angle between 0∘ 0^\circ and 360∘ 360^\circ .
2. Determine the reference angle and quadrant for the equivalent angle.
3. Calculate the sine and cosine of the equivalent angle.
4. Use sine and cosine to find the remaining trigonometric ratios.

STEP 3

Reduce 585∘ 585^\circ by subtracting multiples of 360∘ 360^\circ to find an equivalent angle:
585∘−360∘=225∘ 585^\circ - 360^\circ = 225^\circ
The equivalent angle is 225∘ 225^\circ .

STEP 4

Determine the reference angle and quadrant for 225∘ 225^\circ :
- 225∘ 225^\circ is in the third quadrant. - The reference angle is 225∘−180∘=45∘ 225^\circ - 180^\circ = 45^\circ .

STEP 5

Calculate the sine and cosine of 225∘ 225^\circ using the reference angle 45∘ 45^\circ :
- In the third quadrant, both sine and cosine are negative. - sin⁡(225∘)=−sin⁡(45∘)=−22\sin(225^\circ) = -\sin(45^\circ) = -\frac{\sqrt{2}}{2} - cos⁡(225∘)=−cos⁡(45∘)=−22\cos(225^\circ) = -\cos(45^\circ) = -\frac{\sqrt{2}}{2}

STEP 6

Calculate the remaining trigonometric ratios:
- tan⁡(225∘)=sin⁡(225∘)cos⁡(225∘)=−22−22=1\tan(225^\circ) = \frac{\sin(225^\circ)}{\cos(225^\circ)} = \frac{-\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = 1 - csc⁡(225∘)=1sin⁡(225∘)=−2\csc(225^\circ) = \frac{1}{\sin(225^\circ)} = -\sqrt{2} - sec⁡(225∘)=1cos⁡(225∘)=−2\sec(225^\circ) = \frac{1}{\cos(225^\circ)} = -\sqrt{2} - cot⁡(225∘)=1tan⁡(225∘)=1\cot(225^\circ) = \frac{1}{\tan(225^\circ)} = 1
The exact values of the six trigonometric ratios for 585∘ 585^\circ are:
sin⁡(585∘)=−22,cos⁡(585∘)=−22,tan⁡(585∘)=1\sin(585^\circ) = -\frac{\sqrt{2}}{2}, \quad \cos(585^\circ) = -\frac{\sqrt{2}}{2}, \quad \tan(585^\circ) = 1
csc⁡(585∘)=−2,sec⁡(585∘)=−2,cot⁡(585∘)=1\csc(585^\circ) = -\sqrt{2}, \quad \sec(585^\circ) = -\sqrt{2}, \quad \cot(585^\circ) = 1

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