Question

valuate the expression under the given conditions. tan(θ + ϕ); cos θ = − 1/3 , θ in Quadrant III, sin ϕ = 1/4 , ϕ in Quadrant II

Answer 1

To evaluate tan(θ + ϕ), we use the angle sum formula and identities for sine and cosine to find the individual tangents. We calculate tanθ and tanϕ based on given sine and cosine values, and then substitute them into the angle sum formula to find tan(θ + ϕ).

Step-by-step explanation:

To evaluate the expression tan(θ + ϕ) given cos θ = − 1/3 and sin ϕ = 1/4, we need to use the angle sum formula for tangent:

tan(θ + ϕ) = \frac{tanθ + tanϕ}{1 - tanθtanϕ}

Since we know that cos θ = − 1/3 and θ is in Quadrant III, we can infer that sin θ is negative as well. Using the Pythagorean identity sin²θ + cos²θ = 1, we can find sin θ:

sin θ = -\sqrt{1 - cos²θ} = -\sqrt{1 - (− 1/3)²} = -\sqrt{1 - 1/9} = -\sqrt{8/9} = -\frac{2\sqrt{2}}{3}

Now, since sin ϕ = 1/4 and ϕ is in Quadrant II, we know that cos ϕ is negative. We can find cos ϕ similarly:

cos ϕ = -\sqrt{1 - sin²ϕ} = -\sqrt{1 - (1/4)²} = -\sqrt{1 - 1/16} = -\sqrt{15/16} = -\frac{\sqrt{15}}{4}

Therefore, we can find tan θ and tan ϕ:

tanθ = sinθ / cosθ

tanϕ = sinϕ / cosϕ

Substituting the values we have:

tanθ = -\frac{2\sqrt{2}}{3} / -\frac{1}{3} = 2\sqrt{2}

tanϕ = \frac{1}{4} / -\frac{\sqrt{15}}{4} = -\frac{1}{\sqrt{15}}

Now, we can use these to evaluate tan(θ + ϕ):

tan(θ + ϕ) = \frac{2\sqrt{2} - \frac{1}{\sqrt{15}}}{1 - 2\sqrt{2}(-\frac{1}{\sqrt{15}})} = \frac{2\sqrt{2}\sqrt{15} - 1}{\sqrt{15} - 2\sqrt{30}}

This can be simplified by rationalizing the denominator, but since the calculations will become more complex, it is not necessary for an evaluation unless specifically requested by the student.

Answer 2

To evaluate the expression tan(θ + ϕ) given the conditions cos θ = -1/3 and sin ϕ = 1/4, we can use trigonometric identities. By substituting the known values for cos θ and sin ϕ, we can find sin θ and cos ϕ. Then, we can use the sum formula for tangent to evaluate the expression.

Step-by-step explanation:

To evaluate the expression tan(θ + ϕ) given the conditions cos θ = -1/3 and sin ϕ = 1/4, we need to use the trigonometric identities. First, we can find sin θ using the Pythagorean identity sin²θ + cos²θ = 1. Substituting the value of cos θ, we get sin²θ + (-1/3)² = 1. Solving for sin θ, we find that sin θ = -2√2/3. Next, we can find cos ϕ using the Pythagorean identity sin²ϕ + cos²ϕ = 1. Substituting the value of sin ϕ, we get (1/4)² + cos²ϕ = 1. Solving for cos ϕ, we find that cos ϕ = -15/16.

Now, we can evaluate the expression tan(θ + ϕ) using the sum formula for tangent: tan(θ + ϕ) = (tan θ + tan ϕ)/(1 - tan θ * tan ϕ). Substituting the values, we get:

tan(θ + ϕ) = (sin θ/cos θ + sin ϕ/cos ϕ)/(1 - (sin θ/cos θ) * (sin ϕ/cos ϕ))
= (-2√2/3 + 1/4)/[1 - (-2√2/3) * (1/4)]
= (-8√2 + 3)/(12 - 2√2)

So, the expression tan(θ + ϕ) evaluates to (-8√2 + 3)/(12 - 2√2).