Find the values of the six trigonometric functions for angle θ in standard position if a point with the coordinates (-3,-5) lies on its terminal side.
In this case:
x = - 3 , y = - 5
r = √ ( x² + y² )
r = √ [ ( - 3 )² + (- 5 )² ]
r = √ ( 9 + 25 )
r = √34
sin θ = y / r = - 5 / √34
cos θ = x / r = - 2 / √34
sec θ = 1 / cos θ = - √34 / 2
csc θ = 1 / sin θ = - √34 / 5
tan θ = sin θ / cos θ = ( - 5 / √34 ) / ( - 2 / √34 ) = 5 / 2
cot θ = cos θ / sin θ = ( - 2 / √34 ) / ( - 5 / √34 ) = 2 / 5
To find the values of the six trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) for angle θ in standard position, we need to determine the lengths of the sides of the right triangle formed by the point with coordinates (-3, -5) on its terminal side.
1. First, plot the point (-3, -5) on the coordinate plane.
2. The distance from the origin to this point (-3, -5) is the hypotenuse of the right triangle. Use the Pythagorean theorem to find it: c² = a² + b², where c is the hypotenuse and a and b are the triangle's legs. In this case, a = -3 and b = -5. Thus, c = √((-3)² + (-5)²) = √(9 + 25) = √34.
Now, we can calculate the trigonometric functions:
3. Sine (sin): sin(θ) = opposite/hypotenuse. In this case, the opposite side is -5 and the hypotenuse is √34. So, sin(θ) = -5/√34.
4. Cosine (cos): cos(θ) = adjacent/hypotenuse. Since we have the coordinates of a point on the terminal side of angle θ, we can use the x-coordinate (-3) as the adjacent side. Thus, cos(θ) = -3/√34.
5. Tangent (tan): tan(θ) = sin(θ)/cos(θ). From steps 3 and 4, we have sin(θ) = -5/√34 and cos(θ) = -3/√34. Substituting these values into the tangent formula, we get tan(θ) = (-5/√34)/(-3/√34) = 5/3.
6. Cosecant (csc): csc(θ) = 1/sin(θ). From step 3, sin(θ) = -5/√34. Therefore, csc(θ) = 1/(-5/√34) = -√34/5.
7. Secant (sec): sec(θ) = 1/cos(θ). From step 4, cos(θ) = -3/√34. Thus, sec(θ) = 1/(-3/√34) = -√34/3.
8. Cotangent (cot): cot(θ) = 1/tan(θ). From step 5, tan(θ) = 5/3. Therefore, cot(θ) = 1/(5/3) = 3/5.
In summary:
sin(θ) = -5/√34
cos(θ) = -3/√34
tan(θ) = 5/3
csc(θ) = -√34/5
sec(θ) = -√34/3
cot(θ) = 3/5
To find the values of the six trigonometric functions for angle θ in standard position, we need to determine the lengths of the sides of the triangle formed by the given point (-3, -5) on the terminal side.
Let's denote the length of the hypotenuse as r and the lengths of the other two sides as x and y. Based on the given point (-3, -5), we can calculate these lengths as follows:
x = -3
y = -5
Using the Pythagorean theorem, we can calculate the length of the hypotenuse r:
r = sqrt(x^2 + y^2)
= sqrt((-3)^2 + (-5)^2)
= sqrt(9 + 25)
= sqrt(34)
Now we can find the values of the six trigonometric functions:
1. Sine (sin θ) = y / r = -5 / sqrt(34)
2. Cosine (cos θ) = x / r = -3 / sqrt(34)
3. Tangent (tan θ) = y / x = -5 / -3 = 5/3
4. Cosecant (csc θ) = 1 / sin θ = sqrt(34) / -5
5. Secant (sec θ) = 1 / cos θ = sqrt(34) / -3
6. Cotangent (cot θ) = 1 / tan θ = 3/5