Trig questions, sin cos tan?

Question

im stuck on a few problems, and im not sure how to do it.

Find the exact value of these, no decimal form.
cot(225)
csc(-7pi/4)
(csc42)(sec48)-(tan48)(cot42)
sec(-2pi)

please help out
I have one of them done, cos225 = -sqroot2/2

Thanks

Answer 1

Learn your UNIT CIRCLE!!!

a) 225 - 180 = 45

cot 45 = cos/sin = -sq2/2 * -2/sq2 = 1

b) -7pi-4 = -315 degrees = 45

csc = 1/(-sq2/2) = -2/sq2

c) That one does not seem to make sense.

d) -2pi = -360 = 1

sec 1 = 1 / 1

------ > =1

Answer 2

csc (-7 pi/4) = 1/ sin (-7/4) pi

-7/4 = -1 3/4 The answer to this will lie in the first quadrant. You figure out what the related angle will be.

(csc42)(sec48)-(tan48)(cot42) = (1/sin 42)(1/cos 48) - (sin 48/cos 48)(cos 42/sin 42)

Just find the sine and cosine of these angles. Then do the division, multiplication and subtraction, and you will have the answers.

sec(-2pi) = 1/cos (-2 pi) = 1/cos (0) See below.

cos (-2 pi) = cos (2 pi) = cos (0)

Answer 3

1) Let's convert cot(225) into radian form. All we have to do is multiply 225 by pi/180. This will give us 225pi/180, which reduces to 5pi/4.

Fortunately for us, 5pi/4 is one of our known values on the unit circle.
cot(5pi/4) = cos(5pi/4) / sin(5pi/4) = [-sqrt(2)/2] / [-sqrt(2)/2] = 1

2) To calculate csc(-7pi/4), remember that going a revolution around the unit circle (2pi) brings you at the same place. All you have to do is add 2pi.

csc(-7pi/4) = csc(-7pi/4 + 2pi) = csc(-7pi/4 + 8pi/4) = csc(pi/4)

pi4 is one of our known values, so

csc(pi/4) = 1/sin(pi/4) = 1/[sqrt(2)/2] = 2/sqrt(2) = sqrt(2)

3. csc(42)sec(48) - tan(48)cot(42)

Change everything to sines and cosines.

[1/sin(42)][1/cos(48)] - [sin(48)/cos(48)][cos(42)/sin(42)]

Merge the fractions by multiplying them out.

1/[sin(42)cos(48)] - [sin(48)cos(42)] / [cos(48)sin(42)]

Now, we have a common denominator. Let's merge these two fractions into one.

[1 - sin(48)cos(42)] / [sin(42)cos(48)]

I'm sorry to say I'm not sure where to go from here.

Edit: Through the other math geniuses on here, turns out that if x and y are complimentary angles (angles that equal 90 degrees or pi/2 radians), then sin(x) = cos(y). Therefore,

sin(48) = cos(42). Plugging in sin(48) for every occurrance of cos(42) in the numerator, and doing the opposite for the denominator:

[1 - sin(48)sin(48)] / [cos(48)cos(48)]

[1 - sin^2(48)] / [cos^2(48)]

Note the identity in the numerator; 1 - sin^2(x) = cos^2(x).

[cos^2(48)]/cos^2(48)] = 1

4) To solve sec(-2pi), like we did in question 2, we go one revolution and add 2pi.

sec(-2pi) = sec(-2pi + 2pi) = sec(0) = 1/cos(0) = 1/1 = 1

Answer 4

You seem to be switching back and forth between degrees and radians from problem to problem. Furthermore, I note that all of them except one appear to be multiples of 45° = π/4 radians or even multiples of 90° = π/2 radians. The main thing then is to determine whether the value is positive or negative.

a) cot 225° = cos 225°/sin 225° = (-1/√2)/(-1/√2) = 1

b) csc(-7π/4) = csc(π/4) = 1/sin(π/4) = 1/(1/√2) = √2

c) (csc 42°)(sec 48°) - (tan 48°)(cot 42°)

For this one, first convert everything to sine and cosine. Then, as Scarlet Manuka points out, note that sin 42° = cos 48° as they are complementary angles.

(csc 42°)(sec 48°) - (tan 48°)(cot 42°)
= (1/sin 42°)(1/cos 48°) - (sin 48°/cos 48°)(cos 42°/sin 42°)
= (1/sin 42°)(1/sin 42°) - (sin 48°/sin 42°)(sin 48°/sin 42°)
= (1/sin² 42°) - (sin² 48°/sin² 42°)
= (1 - sin² 48°)/sin² 42°
= (1 - cos² 42°)/sin² 42°
= sin² 42°)/sin² 42°
= 1

d) sec(-2π) = sec(0) = 1/cos(0) = 1/1 = 1

Answer 5

a) cot 225° = cot 45° = 1/tan 45° = 1/1 = 1.
(Remember cot and tan have a period of 180°.)

b) csc (-7π/4) = csc (2π - 7π/4) = csc (π/4) = 1/sin (π/4) = 1/(1/sqrt(2)) = sqrt(2)

c) (csc 42°)(sec 48°) - (tan 48°)(cot 42°)
= (1/sin 42°)(1/cos 48°) - (sin 48°/cos 48°)(cos 42°/sin 42°)
= (1/sin 42°)(1/sin 42°) - (cos 42°/sin 42°)(cos 42°/sin 42°)
(since cos 48° = sin 42° and sin 48° = cos 42° - note that 48° + 42° = 90°)
= 1/sin^2 42° - cos^2 42°/sin^2 42°
= (1 - cos^2 42°) / (sin^2 42°)
= sin^2 42°/sin^2 42°
= 1.

d) sec(-2π) = 1/cos(-2π) = 1/cos(0) = 1/1 = 1.

Answer 6

ok i could desire to know if this could nicely be an actual perspective triangle or no longer? respond and that i will respond to ok considering that there's an actual perspective, then you certainly can use cos rule, so cos(40) = adjacent / hypotenuse so in case you do hypotenuse * cos(40) = adjacent so which you presently fill interior the numbers sixty seven * cos(40) = adjacent, (now because of the fact i'm lazy i cant be hassled with getting off my settee to get the calculator, you you're able to desire to do a sprint little bit of calculations here)

Answer 7

cotangent of 225? Are you sure you're not pulling my leg?