Question

Evaluate the expression under the given conditions. sin(θ − ϕ); tan(θ) = 4/3 , θ in Quadrant III, sin(ϕ) = −3sqrt10/10 , ϕ in Quadrant IV. Give exact value

Answer 1

The expression sin(θ − φ) is evaluated using the given values for tan(θ) and sin(φ) along with the trigonometric identity for sin(a − b). Both angles are in quadrants where the sine and cosine take on specific signs. The exact value is calculated by determining the corresponding sine and cosine for each angle and applying the identity.

Step-by-step explanation:

To evaluate the expression sin(θ − φ), given tan(θ) = 4/3, with θ in Quadrant III, and sin(φ) = −3√10/10, with φ in Quadrant IV, we can use the trigonometric identity sin(a − b) = sin(a)cos(b) − cos(a)sin(b).

Since tan(θ) = 4/3, we can represent sin(θ) and cos(θ) as the lengths of the sides of a right triangle where the opposite side is 4, the adjacent side is 3, and the hypotenuse h is found using the Pythagorean theorem: h = √(4^2 + 3^2) = √25 = 5. However, because θ is in the third quadrant where sine is negative and cosine is also negative, we have sin(θ) = −4/5 and cos(θ) = −3/5.

For φ, since we only have sin(φ), we can find cos(φ) by recognizing that cos^2(φ) = 1 − sin^2(φ) = 1 − (−3√10/10)^2. Since φ is in the fourth quadrant, where sine is negative and cosine is positive, we have cos(φ) = √(1 − 0.9) = √0.1.

Now we can plug these values into our formula: sin(θ − φ) = sin(θ)cos(φ) − cos(θ)sin(φ) = (−4/5)√0.1 − (−3/5)(-3√10/10). With some arithmetic, we find the exact value of sin(θ − φ).

Answer 2

To find sin(θ - φ) with given trigonometric ratios and quadrant information, we used the sine difference identity and properties of the trigonometric functions in specific quadrants to obtain the exact value of -13√10/50.

Step-by-step explanation:

We are tasked with evaluating sin(θ - φ) given the conditions: tan(θ) = 4/3, where θ is in the third quadrant, and sin(φ) = -3√10/10, where φ is in the fourth quadrant. To do this, we will use trigonometric identities and properties of each quadrant.

Since tan(θ) = 4/3, we can represent the opposite and adjacent sides of θ using these values in a right triangle. In the third quadrant, both sine and cosine are negative, thus sin(θ) is -4/5 and cos(θ) is -3/5. For φ, we have sin(φ) = -3√10/10 and since φ is in the fourth quadrant where cosine is positive, cos(φ) can be found using the Pythagorean theorem resulting in cos(φ) = 1/√10.

Using the sine difference identity sin(α - β) = sin(α)cos(β) - cos(α)sin(β), we substitute the values to get sin(θ - φ) = (-4/5)(1/√10) - (-3/5)(-3√10/10) = -4√10/50 - 9√10/50 = -13√10/50 = -13√10/50. This is the exact value of sin(θ - φ).