Given tan Θ = - 8/5 and sin Θ < 0, find sin Θ, cos Θ, csc Θ, sec Θ and cot Θ.
since tangent is negative in II and IV and the sine is negative in III and IV , our angle Ø must be in quadrant IV
so using our given tanØ = -8/5
y = -8, x = 5
and r^2 = 64+25 = 89
r = √89
sinØ = y/r = -8/√89
cosØ = x/r = 5/√89
cscØ = -√89/8
secØ = √89/5
cotØ = -5/8
Given: tan Θ = -8/5 and sin Θ < 0.
Since tan Θ = -8/5, we can use the Pythagorean identity to find the value of cos Θ:
tan Θ = sin Θ / cos Θ
-8/5 = sin Θ / cos Θ
Next, we need to find sin Θ. We know that sin Θ < 0, which means it is negative. Since sin Θ is negative, we can set it to -1.
sin Θ = -1
Now we can substitute this value into the equation for tan Θ:
-1 / cos Θ = -8/5
To solve for cos Θ, we can cross-multiply and solve for cos Θ:
-1 * 5 = -8 * cos Θ
-5 = -8 cos Θ
Dividing both sides by -8 gives:
cos Θ = 5/8
Now we can use the values of sin Θ and cos Θ to find the remaining trigonometric ratios:
csc Θ = 1/sin Θ
csc Θ = 1/(-1)
csc Θ = -1
sec Θ = 1/cos Θ
sec Θ = 1/(5/8)
sec Θ = 8/5
cot Θ = cos Θ / sin Θ
cot Θ = (5/8) / (-1)
cot Θ = -5/8
So, the values of the trigonometric ratios are:
sin Θ = -1
cos Θ = 5/8
csc Θ = -1
sec Θ = 8/5
cot Θ = -5/8
To find the values of sin Θ, cos Θ, csc Θ, sec Θ, and cot Θ, given that tan Θ = -8/5 and sin Θ < 0, we'll use the given information and some trigonometric identities.
We know that:
1. tan Θ = sin Θ / cos Θ
2. csc Θ = 1 / sin Θ
3. sec Θ = 1 / cos Θ
4. cot Θ = 1 / tan Θ
First, let's find cos Θ using the given information tan Θ = -8/5:
Using the identity tan Θ = sin Θ / cos Θ, we can rearrange it to cos Θ = sin Θ / tan Θ.
Let's substitute the given values:
cos Θ = sin Θ / (-8/5) = (5/8) * sin Θ
Since sin Θ < 0, we can also determine the sign of cos Θ. Since cos Θ = sin Θ / (-8/5), if sin Θ is negative, then cos Θ will also be negative.
Now, let's find sin Θ:
Using the given information tan Θ = -8/5, we can use the Pythagorean identity:
tan^2 Θ + 1 = sec^2 Θ
(-8/5)^2 + 1 = sec^2 Θ
64/25 + 1 = sec^2 Θ
89/25 = sec^2 Θ
Since sec^2 Θ = 1 / cos^2 Θ, we can solve for cos Θ:
cos^2 Θ = 25/89
cos Θ = ± √(25/89)
Since we already determined that cos Θ is negative, we can use the negative value of √(25/89).
Now, with cos Θ and sin Θ known, we can find the values of the remaining trigonometric functions:
1. sin Θ: Since we know the sign of sin Θ is negative, we can use the formula sin^2 Θ + cos^2 Θ = 1 to calculate sin Θ:
sin^2 Θ + (25/89) = 1
sin^2 Θ = 64/89
sin Θ = -√(64/89) = -8/√89
2. csc Θ: Using the formula csc Θ = 1 / sin Θ:
csc Θ = 1 / (-8/√89) = -√89 / 8
3. sec Θ: We already found that sec^2 Θ = 89/25, so:
sec Θ = ± √(89/25)
Since we already determined that sec Θ is negative, we can use the negative value of √(89/25).
4. cot Θ: Using the formula cot Θ = 1 / tan Θ:
cot Θ = 1 / (-8/5) = -5/8
So, the values of the trigonometric functions are:
sin Θ = -8/√89
cos Θ = -√(25/89)
csc Θ = -√89 / 8
sec Θ = -√(89/25)
cot Θ = -5/8