Suppose ABC is a right triangle with sides of lengths a, b, and c and right angle at C. Find the unknown side length using the Pythagorean Theorem and find the following trigonometric functions of the indicated angle.
Given: a = 4 and b = 7
Find: sin A, cot B, sec A, tan B
sinA = a/c = cosB
cosA = b/c = sinB
tanA = a/b = cotB
plug and chug.
To find the unknown side length using the Pythagorean Theorem, we can use the equation a^2 + b^2 = c^2. Given that a = 4 and b = 7, we can substitute these values into the equation to solve for c.
Using the Pythagorean Theorem:
4^2 + 7^2 = c^2
16 + 49 = c^2
65 = c^2
To find c, we take the square root of both sides:
√65 = c
c ≈ 8.06
Therefore, the unknown side length (c) is approximately 8.06.
Now, let's move on to finding the trigonometric functions of the indicated angles.
1. sin A:
To find sin A, we need to use the opposite side (a) and the hypotenuse (c) of angle A. Therefore, sin A = a/c.
sin A = 4/8.06
sin A ≈ 0.496
Hence, sin A is approximately 0.496.
2. cot B:
To find cot B, we use the adjacent side (b) and the opposite side (a) of angle B. Therefore, cot B = b/a.
cot B = 7/4
cot B = 1.75
So, cot B is equal to 1.75.
3. sec A:
Secant is the reciprocal of cosine. To find sec A, we need to use the adjacent side (b) and the hypotenuse (c) of angle A. Therefore, sec A = c/b.
sec A = 8.06/7
sec A ≈ 1.151
Therefore, sec A is approximately 1.151.
4. tan B:
To find tan B, we use the opposite side (a) and the adjacent side (b) of angle B. Therefore, tan B = a/b.
tan B = 4/7
tan B ≈ 0.571
Hence, tan B is approximately 0.571.
In summary:
sin A ≈ 0.496
cot B = 1.75
sec A ≈ 1.151
tan B ≈ 0.571
To find the unknown side length using the Pythagorean theorem, we can use the formula:
c^2 = a^2 + b^2
Given that a = 4 and b = 7, we can substitute these values into the formula:
c^2 = 4^2 + 7^2
c^2 = 16 + 49
c^2 = 65
Taking the square root of both sides, we find:
c = √65
Now let's find the trigonometric functions of the indicated angles.
1. To find sin A, we can use the formula:
sin A = opposite / hypotenuse
In this case, the opposite side to angle A is side b (with length 7) and the hypotenuse is side c (with length √65). Substituting these values into the formula, we get:
sin A = 7 / √65
2. To find cot B, we can use the formula:
cot B = adjacent / opposite
In this case, the adjacent side to angle B is side a (with length 4) and the opposite side is side b (with length 7). Substituting these values into the formula, we get:
cot B = 4 / 7
3. To find sec A, we can use the formula:
sec A = hypotenuse / adjacent
In this case, the hypotenuse is side c (with length √65) and the adjacent side to angle A is side a (with length 4). Substituting these values into the formula, we get:
sec A = √65 / 4
4. To find tan B, we can use the formula:
tan B = opposite / adjacent
In this case, the opposite side to angle B is side b (with length 7) and the adjacent side is side a (with length 4). Substituting these values into the formula, we get:
tan B = 7 / 4
Therefore, sin A = 7 / √65, cot B = 4 / 7, sec A = √65 / 4, and tan B = 7 / 4.