1. θ is a first quadrant angle in standard position and cos 1/(sqrt10)= . Find the exact value of sin .
2. feta is a second quadrant angle in standard position and tan(feta) -((sqrt5)/2) = . Find the exact value of cos(feta) .
3. θ is an angle in standard position with domain 0° <θ< 360° and cot θ -(24/7)= . Find all possible values of θ (accurate to 0.1° if necessary)
feta = β
1. draw a diagram. Legs are 1 and 3, hypotenuse is √10. If cosθ = 1/√10, sinθ = 3/√10
2. legs are √5 and 2, hypotenuse = 3. If tanθ = √5/2, cosθ = 2/3. In QII, cosθ = -2/3
3. legs are 7,24, hypotenuse =25
if cotθ < 0, θ is QII or QIV.
Arccot(24/7) = 16.26°, so
θ = 163.74° or 343.74°
Sure! Let's solve each of the problems step by step:
1. To find the exact value of sinθ given that cosθ = 1/(sqrt10), we can use the Pythagorean identity sin²θ + cos²θ = 1. Since we know the value of cosθ, we can substitute it into the equation and solve for sinθ.
Let's begin:
cos²θ = 1/(sqrt10)²
cos²θ = 1/10
Now, using the Pythagorean identity, we can write:
sin²θ = 1 - cos²θ
sin²θ = 1 - 1/10
sin²θ = 9/10
To find the exact value of sinθ, we take the square root of both sides of the equation:
sinθ = ± √(9/10)
sinθ = ± 3/√10
sinθ = ± (3√10) / 10
Therefore, the exact value of sinθ is ± (3√10) / 10.
2. To find the exact value of cos(feta) given that tan(feta) - ((sqrt5)/2) = 0, we need to first find the value of tan(feta) and then use it to determine cos(feta).
Let's solve for tan(feta):
tan(feta) = ((sqrt5)/2)
Since we're working in the second quadrant, tan(feta) is positive and sin(feta) is also positive. Therefore, cos(feta) will be negative.
Now, let's find sin(feta) using the Pythagorean identity sin²(feta) + cos²(feta) = 1:
sin²(feta) = 1 - cos²(feta)
sin²(feta) = 1 - cos²(feta)
sin²(feta) = 1 - (cos(feta))²
sin(feta) = √(1 - (cos(feta))²)
sin(feta) = √(1 - ((sqrt5)/2)²)
sin(feta) = √(1 - 5/4)
sin(feta) = √(4/4 - 5/4)
sin(feta) = √(-1/4)
sin(feta) = (i/2)
Since cos(feta) is negative in the second quadrant, we can write:
cos(feta) = -√(1 - sin²(feta))
cos(feta) = -√(1 - (i/2)²)
cos(feta) = -√(1 - (-1/4))
cos(feta) = -√(5/4)
cos(feta) = -(√5)/2
Therefore, the exact value of cos(feta) is -(√5)/2.
3. To find all possible values of θ when cot(θ) = -24/7, we need to find the angles that satisfy this condition within the given domain of 0° < θ < 360°.
cot(θ) = -24/7 is equivalent to tan(θ) = -7/24.
Since tangent is negative in the second and fourth quadrants, we need to find the angles within those quadrants that have a tangent of -7/24.
Using an inverse trigonometric function, we can find the principal angle in the second quadrant with a tangent of -7/24:
θ = arctan(-7/24)
Using a calculator, we find that θ ≈ -17.89° (accurate to 0.1°).
Now, to find the corresponding angle in the fourth quadrant, we can subtract the principal angle from 360°:
θ = 360° - 17.89°
θ ≈ 342.11° (accurate to 0.1°)
Therefore, the possible values of θ (accurate to 0.1°) when cot(θ) = -24/7 within the given domain are approximately -17.89° and 342.11°.