So the problem is asking to find cos(α+β)
The problem gives me
sinα= 12/13, 0<α<π/2
cosβ= 21/29, 0<β<π/2
The answer is -135/377
I have no idea how to start solving to find that answer.
For each of the given trig ratios, make a sketch of the corresponding right-angled triangles
sinα = 12/13 , so y = 12, r = 13
x^2 + y^2 = r^2
x^2 + 144 = 169
x^2 = 25
x = 5 in quad I , so
sinα = 12/13 , cosα = 5/13
cosβ = 21/29 in quadrant I
x = 21, r=29
y = 20
sinβ = 20/29 , cosβ = 21/29
You should have learned that
cos(α+β) = cosαcosβ - sinαsinβ
= (5/13)(21/29)- (12/13)(20/29)
= -135/377
Thank you ^^
To find the value of cos(α + β), we need to use the trigonometric identities and the given information about sinα and cosβ.
First, let's visualize the situation using the right triangle. We know that sinα = 12/13, which means that the side opposite angle α is 12, and the hypotenuse is 13. Similarly, cosβ = 21/29, so the adjacent side to angle β is 21, and the hypotenuse is 29.
Let's label the sides of the triangle:
Opposite angle α = 12
Adjacent to angle β = 21
Hypotenuse angle α = 13
Hypotenuse angle β = 29
Since we need to find cos(α + β), one approach is to use the cosine of the sum formula:
cos(α + β) = cosα * cosβ - sinα * sinβ
Now we substitute the known values into the formula:
cos(α + β) = (12/13) * (21/29) - (12/13) * sinβ
Now we need to find sinβ. To do that, we can use the Pythagorean theorem:
sinβ = √(1 - cos²β)
Using the value of cosβ (21/29):
sinβ = √(1 - (21/29)²)
Now we can calculate sinβ:
sinβ = √(1 - (441/841))
= √(841/841 - 441/841)
= √(400/841)
= 20/29
Now we can substitute sinβ into our original equation:
cos(α + β) = (12/13) * (21/29) - (12/13) * (20/29)
Simplifying this expression:
cos(α + β) = (252/377) - (240/377)
= 12/377
Therefore, cos(α + β) = 12/377, not -135/377. Please double-check your answer, as it appears to be incorrect.