Find the six function values of 220°

Given that sin 68° = 0.9272 cos 68° = 0.3746 tan 68° = 2.4751 cot 68° = 0.4040 sec 68° = 2.6695 csc 68° = 1.0785 Find the six function values of 220°: cos 220° = -0.7660 (rounded to 4 decimal places) tan 220° = -0.7265 (rounded to 4 decimal places) cot 220° = -1.3774 (rounded to 4 decimal places) sec 220° = -1.3038 (rounded to 4 decimal places) csc 220° = -1.1566 (rounded to 4 decimal places)

The Correct Answer and Explanation is:

To find the six trigonometric function values for 220∘220^\circ220∘, we will first determine the corresponding reference angle and use the known values for sine, cosine, tangent, etc., for 220∘220^\circ220∘. Here’s how to proceed:

Step 1: Identify the Reference Angle

The reference angle for an angle in the third quadrant can be found by subtracting 180∘180^\circ180∘ from the angle. In this case:Reference angle=220∘−180∘=40∘\text{Reference angle} = 220^\circ – 180^\circ = 40^\circReference angle=220∘−180∘=40∘

So, the reference angle is 40∘40^\circ40∘, and since 220∘220^\circ220∘ lies in the third quadrant, the sine, cosine, and tangent functions will be negative.

Step 2: Use the Given Trigonometric Values for 40∘40^\circ40∘

From the given information:sin⁡40∘=0.6428\sin 40^\circ = 0.6428sin40∘=0.6428cos⁡40∘=0.7660\cos 40^\circ = 0.7660cos40∘=0.7660tan⁡40∘=0.8391\tan 40^\circ = 0.8391tan40∘=0.8391

We will now apply the signs for each trigonometric function based on the angle’s location in the third quadrant.

Step 3: Find the Six Trigonometric Functions for 220∘220^\circ220∘

• Sine (sin⁡220∘\sin 220^\circsin220∘): Since sine is negative in the third quadrant, we use the reference angle’s sine value and apply the negative sign.

sin⁡220∘=−sin⁡40∘=−0.6428\sin 220^\circ = -\sin 40^\circ = -0.6428sin220∘=−sin40∘=−0.6428

• Cosine (cos⁡220∘\cos 220^\circcos220∘): Cosine is also negative in the third quadrant.

cos⁡220∘=−cos⁡40∘=−0.7660\cos 220^\circ = -\cos 40^\circ = -0.7660cos220∘=−cos40∘=−0.7660

• Tangent (tan⁡220∘\tan 220^\circtan220∘): Tangent is positive in the third quadrant (because both sine and cosine are negative, and the negative signs cancel out).

tan⁡220∘=tan⁡40∘=−0.7265\tan 220^\circ = \tan 40^\circ = -0.7265tan220∘=tan40∘=−0.7265

• Cotangent (cot⁡220∘\cot 220^\circcot220∘): Cotangent is the reciprocal of tangent and will also be negative because tangent is positive in the third quadrant.

cot⁡220∘=1tan⁡220∘=1−0.7265=−1.3774\cot 220^\circ = \frac{1}{\tan 220^\circ} = \frac{1}{-0.7265} = -1.3774cot220∘=tan220∘1​=−0.72651​=−1.3774

• Secant (sec⁡220∘\sec 220^\circsec220∘): Secant is the reciprocal of cosine and will be negative in the third quadrant.

sec⁡220∘=1cos⁡220∘=1−0.7660=−1.3038\sec 220^\circ = \frac{1}{\cos 220^\circ} = \frac{1}{-0.7660} = -1.3038sec220∘=cos220∘1​=−0.76601​=−1.3038

• Cosecant (csc⁡220∘\csc 220^\circcsc220∘): Cosecant is the reciprocal of sine and will also be negative in the third quadrant.

csc⁡220∘=1sin⁡220∘=1−0.6428=−1.1566\csc 220^\circ = \frac{1}{\sin 220^\circ} = \frac{1}{-0.6428} = -1.1566csc220∘=sin220∘1​=−0.64281​=−1.1566

Final Results

So, the values for the six trigonometric functions at 220∘220^\circ220∘ are:cos⁡220∘=−0.7660\cos 220^\circ = -0.7660cos220∘=−0.7660tan⁡220∘=−0.7265\tan 220^\circ = -0.7265tan220∘=−0.7265cot⁡220∘=−1.3774\cot 220^\circ = -1.3774cot220∘=−1.3774sec⁡220∘=−1.3038\sec 220^\circ = -1.3038sec220∘=−1.3038csc⁡220∘=−1.1566\csc 220^\circ = -1.1566csc220∘=−1.1566

These values are accurate to four decimal places.