Find the values of the trigonometry function of t from the given information tan(t)=1/5 terminal point of t is in quadrant 3
Then sin(t) cost(t) csc (t) sec(t) cot(t)
To find the values of the trigonometric functions for t, we can use the given information that tan(t) = 1/5 and t is in quadrant 3. In quadrant 3, both the sine and cosine functions are negative.
We can start by finding the values of sine and cosine using the tangent:
tan(t) = 1/5
opposite/adjacent = 1/5
opposite = 1, adjacent = 5
Using the Pythagorean theorem, we can find the hypotenuse:
hypotenuse^2 = opposite^2 + adjacent^2
hypotenuse^2 = 1^2 + 5^2
hypotenuse^2 = 26
hypotenuse = √26
Now we can find the values of sine and cosine:
sin(t) = opposite/hypotenuse = 1/√26 = √26/26 = √26/26
cos(t) = adjacent/hypotenuse = 5/√26 = 5√26/26 = 5√26/26
Next, we can find the values of csc(t), sec(t), and cot(t) using the reciprocal identities:
csc(t) = 1/sin(t) = 1/(√26/26) = 26/√26 = 26√26/26 = √26
sec(t) = 1/cos(t) = 1/(5√26/26) = 26/(5√26) = 26√26/(5√26) = 26/5
cot(t) = 1/tan(t) = 1/(1/5) = 5
So, the values of the trigonometric functions for t are:
sin(t) = √26/26
cos(t) = 5√26/26
csc(t) = √26
sec(t) = 26/5
cot(t) = 5
To find the values of the trigonometric functions of t with the given information, we can use the fact that tangent is equal to the ratio of sine to cosine:
tan(t) = sin(t) / cos(t)
Given that tan(t) = 1/5, we can write the equation as:
1/5 = sin(t) / cos(t)
To solve for sin(t) and cos(t), we can use the Pythagorean identity:
sin^2(t) + cos^2(t) = 1
Since the terminal point of t is in quadrant 3, we know that cosine is negative in that quadrant. Therefore, cos(t) = -sqrt(1 - sin^2(t)).
Substituting the value of cos(t) in the tangent equation, we get:
1/5 = sin(t) / (-sqrt(1 - sin^2(t)))
Cross-multiplying, we have:
-5sin(t) = sqrt(1 - sin^2(t))
Squaring both sides, we get:
25sin^2(t) = 1 - sin^2(t)
26sin^2(t) = 1
sin^2(t) = 1 / 26
Taking the square root of both sides, we have:
sin(t) = ± sqrt(1 / 26)
Now, we can find cos(t) by substituting the value of sin(t) in the equation:
cos(t) = -sqrt(1 - sin^2(t))
cos(t) = -sqrt(1 - 1/26)
cos(t) = -sqrt(25 / 26)
Next, let's find the values of the remaining trigonometric functions.
csc(t) = 1 / sin(t)
csc(t) = 1 / (± sqrt(1 / 26))
csc(t) = ± sqrt(26)
sec(t) = 1 / cos(t)
sec(t) = 1 / (-sqrt(25 / 26))
sec(t) = -sqrt(26) / 5
cot(t) = 1 / tan(t)
cot(t) = 1 / (1/5)
cot(t) = 5
Therefore, the values of the trigonometric functions are:
sin(t) = ± sqrt(1 / 26)
cos(t) = -sqrt(25 / 26)
csc(t) = ± sqrt(26)
sec(t) = -sqrt(26) / 5
cot(t) = 5
To find the values of the trigonometric functions sin(t), cos(t), csc(t), sec(t), and cot(t) given that tan(t) = 1/5 and the terminal point of t is in quadrant 3, we can use the following steps:
1. Recall that tan(t) = sin(t) / cos(t). Since tan(t) is 1/5, we have the ratio sin(t) / cos(t) = 1/5.
2. To determine the sign of sin(t) and cos(t) in quadrant 3, we know that sin(t) < 0 and cos(t) < 0 because both sine and cosine are negative in that quadrant.
Therefore, we can rewrite the ratio as sin(t) / cos(t) = -1/5.
Now, we can solve for sin(t) and cos(t) using the trigonometric identity sin^2(t) + cos^2(t) = 1.
3. Square both sides of the equation sin(t) / cos(t) = -1/5 to get (sin(t) / cos(t))^2 = (-1/5)^2.
This simplifies to sin^2(t) / cos^2(t) = 1/25.
4. Rearrange the equation to sin^2(t) = cos^2(t) / 25.
Substituting the identity cos^2(t) = 1 - sin^2(t), we have sin^2(t) = (1 - sin^2(t)) / 25.
5. Multiply both sides of the equation by 25 to eliminate the denominator: 25sin^2(t) = 1 - sin^2(t).
6. Combine like terms: 26sin^2(t) = 1.
7. Divide both sides by 26: sin^2(t) = 1/26.
8. Take the square root of both sides: sin(t) = ±sqrt(1/26).
Since we know sin(t) < 0 in quadrant 3, we take the negative square root: sin(t) = -sqrt(1/26).
9. Now, we can find cos(t) using the equation sin^2(t) + cos^2(t) = 1.
Substituting -sqrt(1/26) for sin(t), we have (-sqrt(1/26))^2 + cos^2(t) = 1.
10. Simplify the equation: 1/26 + cos^2(t) = 1.
11. Subtract 1/26 from both sides: cos^2(t) = 25/26.
12. Take the square root of both sides: cos(t) = ±sqrt(25/26).
Since we know cos(t) < 0 in quadrant 3, we take the negative square root: cos(t) = -sqrt(25/26).
Now, we can calculate csc(t), sec(t), and cot(t) using the reciprocal identities:
csc(t) = 1 / sin(t) = 1 / (-sqrt(1/26)) = -sqrt(26).
sec(t) = 1 / cos(t) = 1 / (-sqrt(25/26)) = -sqrt(26/25).
cot(t) = cos(t) / sin(t) = (-sqrt(25/26)) / (-sqrt(1/26)) = sqrt(26).
Therefore, the values of the trigonometric functions are sin(t) = -sqrt(1/26), cos(t) = -sqrt(25/26), csc(t) = -sqrt(26), sec(t) = -sqrt(26/25), and cot(t) = sqrt(26).