in triangle ABC,right-angled at A,if cot B=1, find the value of cosB cosA+sinB sinC
cotB=1 so B=pi/4, so C=pi/4
cosA = 0
sinB = sinC = 1/√2
. . .
To find the value of cosB cosA + sinB sinC in triangle ABC, right-angled at A, we need to make use of the given information that cot B = 1.
First, let's determine the values of cosB and sinB using the given information. We know that cot B is the reciprocal of tan B, so if cot B = 1, then tan B = 1/1 = 1.
Since the triangle is right-angled at A, we can use the concept of trigonometric ratios to determine the values of cosB and sinB. In a right-angled triangle, the cosine of an angle is equal to the adjacent side divided by the hypotenuse, and the sine of an angle is equal to the opposite side divided by the hypotenuse.
Let's assume that the length of the side opposite angle B is x, and the length of the side adjacent to angle B is y. In this case, tan B = 1, so x/y = 1.
Using the Pythagorean theorem, we know that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, it is given that the triangle is right-angled at A, so A must be our right angle.
Applying the Pythagorean theorem, we have x^2 + y^2 = c^2, where c is the length of the hypotenuse.
Since the triangle is right-angled at A, we can also write sinB = x/c and cosB = y/c.
Let's rearrange these equations:
x/y = 1 --> x = y (Equation 1)
sinB = x/c --> sinB = y/c (Equation 2)
cosB = y/c --> cosB = x/c (Equation 3)
Now, let's substitute Equation 1 into Equations 2 and 3 to find the values of sinB and cosB:
sinB = y/c
= x/c (Substituting x = y from Equation 1)
= cosB
cosB = x/c
= y/c (Substituting x = y from Equation 1)
= sinB
From the above equations, we can conclude that sinB = cosB.
Now, let's find the value of cosA. Since the triangle is right-angled at A, cosA = adjacent side/hypotenuse, which is y/c.
Finally, let's find the value of sinC. Since the triangle is right-angled at A, sinC = opposite side / hypotenuse, which is x/c.
The value we need to find is cosB cosA + sinB sinC, which is essentially cosB * (adjacent side/hypotenuse) + sinB * (opposite side/hypotenuse):
cosB cosA + sinB sinC = cosB * (y/c) + sinB * (x/c)
= cosB * (cosB) + sinB * (sinC)
= (cosB)^2 + (sinB)^2 (since sinC = sinB)
= 1^2 (since sinB = cosB)
= 1
Therefore, the value of cosB cosA + sinB sinC in triangle ABC, right-angled at A, when cot B = 1, is equal to 1.