cotA = tan(90degrees- A) True or False
The vectors <4, 5> and <-10, 8> are orthogonal True or False
2<3, 5> = 16 True or False
<2. 3>*<4, 3> =
<2, 6> + <4, 5> =
Find c if a = 3, b = 5, and A = 30degrees=
To answer these questions, let's break them down one by one:
1. cotA = tan(90 degrees - A)
To determine if this statement is true or false, we need to know the relationship between the cotangent (cot) and tangent (tan) functions. The cotangent of an angle A is equal to the tangent of the complement of A (90 degrees - A). Therefore, the statement is TRUE.
2. The vectors <4, 5> and <-10, 8> are orthogonal
Two vectors are considered orthogonal if their dot product is zero. To find the dot product, multiply the corresponding components of the vectors and sum them together. In this case, we have (4 * -10) + (5 * 8) = -40 + 40 = 0. Since the dot product is zero, the vectors are indeed orthogonal. Therefore, the statement is TRUE.
3. 2<3, 5> = 16
To determine if this statement is true or false, we need to apply scalar multiplication to the vector <3, 5>. Multiply each component of the vector by the scalar 2, resulting in <2 * 3, 2 * 5> = <6, 10>. The statement is FALSE because <6, 10> does not equal 16.
4. <2, 3> * <4, 3> = ?
The multiplication of two vectors is not as straightforward as scalar multiplication. To find the dot product of two vectors, multiply the corresponding components and sum them. Using the provided vectors, we have (2 * 4) + (3 * 3) = 8 + 9 = 17. Therefore, the dot product of <2, 3> and <4, 3> is 17.
5. <2, 6> + <4, 5> = ?
To add two vectors, add their corresponding components. Using the provided vectors, we have (2 + 4, 6 + 5) = <6, 11>.
6. Find c if a = 3, b = 5, and A = 30 degrees
To find the missing side, c, of a triangle with known sides a and b and the included angle A, we can use the Law of Cosines. The formula is as follows: c^2 = a^2 + b^2 - 2ab * cos(A). Plugging in the given values, we have c^2 = 3^2 + 5^2 - 2 * 3 * 5 * cos(30 degrees). Evaluating it further, c^2 = 9 + 25 - 30 * cos(30 degrees). Using the cosine function, cos(30 degrees) = sqrt(3)/2. Substituting this value, c^2 = 9 + 25 - 30 * (sqrt(3)/2). Simplifying, c^2 = 34 - 15 * sqrt(3). Taking the square root of both sides, c = sqrt(34 - 15 * sqrt(3)).