1. Find the exact value of tan660 using the functions of 330deg.
2. Solve the equation 2sin^20+5sin0+3=0 on the interval 0<=0<2pi.
tan 660°
= tan 300° , (660 - 360 - 300)
= - tan 60
= -√3/1 = -√3
2sin^2Ø + 5sinØ + 3 = 0
(2sinØ - 1)(sinØ + 3) = 0
sinØ = 1/2 or sinØ = -3, the latter is not possible
if sinØ = 1/2
then Ø = 30° or Ø = 150°
or
Ø = π/6 or Ø = 5π/6
1 7/2 divided by 7 11/12 =
To find the exact value of tan(660) using the functions of 330°, we can make use of the periodicity and symmetry of trigonometric functions.
1. Convert 660° into an equivalent angle within the range of 0° to 360° using the formula: equivalent angle = angle mod 360°.
In this case, 660° mod 360° = 300°.
2. Since the given functions are only for 330°, we need to find the corresponding angle within that range.
To do this, we can subtract 330° from the equivalent angle: 300° - 330° = -30°.
3. Now, we know that the tangent function is periodic every 180°, so we can determine the angle's position within one period by adding or subtracting multiples of 180°.
In this case, we add 180° to -30° to bring it within the range of 0° to 360°: -30° + 180° = 150°.
4. Finally, we can use the tangent function to find the exact value: tan(150°) = tan(150° - 180°) = tan(-30°).
However, the tangent function has symmetries at 180° and 360°.
So, tan(-30°) is equivalent to tan(-30° + 180°) = tan(150°).
Therefore, the exact value of tan(660) using the functions of 330° is equal to tan(150°).
Now, let's solve the equation 2sin^2θ + 5sinθ + 3 = 0 on the interval 0 ≤ θ < 2π.
1. Notice that this equation is in terms of sine (sinθ) with a quadratic form. Let's try to factor it.
2sin^2θ + 5sinθ + 3 = 0
Factoring the equation, we have:
(2sinθ + 3)(sinθ + 1) = 0
2. To solve this equation, we set each factor equal to zero and solve for θ individually:
2sinθ + 3 = 0 or sinθ + 1 = 0
For 2sinθ + 3 = 0:
2sinθ = -3
sinθ = -3/2
However, sine can only take values between -1 and 1, so there are no solutions for this equation.
For sinθ + 1 = 0:
sinθ = -1
3. To find the solutions for sinθ = -1 on the interval 0 ≤ θ < 2π, we can look for the angles where sine equals -1.
One of these angles is θ = 3π/2 or 270°.
4. Therefore, the equation 2sin^2θ + 5sinθ + 3 = 0 has one solution on the interval 0 ≤ θ < 2π:
θ = 3π/2 or 270°.