If 13 cosP= 5 P¢ã[0,270]and 15 tan Q-8=0 for sinQ<0,evaluate: 17tanP.sinQ
P is in QI, so if cosP = 5/13, tanP=12/5
Q is in QIII, so if tanQ = 8/15, sinQ = -8/17
17tanPsinQ = 17(12/5)(-8/17) = -96/5
To evaluate 17tanP.sinQ, we first need to find the values of P and Q.
We are given that 13cosP = 5P in the range [0, 270] and 15tanQ - 8 = 0 for sinQ < 0.
Let's solve these equations step by step.
1. 13cosP = 5P:
Dividing both sides by 13, we get:
cosP = 5P/13
2. 15tanQ - 8 = 0:
Adding 8 to both sides, we get:
15tanQ = 8
Dividing both sides by 15, we get:
tanQ = 8/15
Now let's find the values of P and Q using trigonometric identities.
1. For cosP = 5P/13:
Since cosP is positive in the range [0, 270], we need to find the value of P between 0 and 270 degrees (inclusive) that satisfies the equation.
One possible value is P = 0 degrees.
2. For tanQ = 8/15:
Since sinQ < 0, we know that Q is in Quadrant III or Quadrant IV.
In Quadrant III, tanQ is negative.
So, let's find the angle in Quadrant III whose tangent gives 8/15.
We can use the inverse tangent function (arctan) to find Q:
Q = arctan(-8/15)
Using a calculator, we find Q ≈ -28.07 degrees.
Now we can evaluate 17tanP.sinQ:
17tanP.sinQ = 17 * tan(0 degrees) * sin(-28.07 degrees)
= 17 * (0) * (-0.4695) [using the values for P and Q]
= 0.
Therefore, the value of 17tanP.sinQ is 0.
To evaluate 17tanP.sinQ, we first need to find the values of P and Q from the given equations.
Let's start with the first equation: 13 cosP = 5 P in the domain [0, 270]. This equation suggests that cosP and P must be related such that their product is equal to 5/13.
To find the value of P, we can use the inverse cosine function. Since the cosine function has a range of -1 to 1, we need to check if 5/13 is within that range. If it is, we can use the inverse cosine directly.
cosP = 5/13
To find P, we take the inverse cosine of both sides:
P = arccos(5/13)
Similarly, let's move on to the second equation: 15 tanQ - 8 = 0 for sinQ < 0.
We can add 8 to both sides of the equation to isolate the term with tanQ:
15 tanQ = 8
Then divide both sides by 15:
tanQ = 8/15
Now we have the values of P and Q.
To evaluate 17tanP.sinQ, we substitute the values of P and Q into the expression:
17tanP.sinQ = 17 * tan(arccos(5/13)) * sin(arctan(8/15))
Using a scientific calculator, we can find the values of the inverse trigonometric functions and then evaluate the expression.