Using trigonometric ratios find the remaining side lengths and angle measurements to 2 decimal places each:
(a) α = 35°, β = 90°, a = 30
(b) α = 90°, b = 7, c = 6
(c) a = 33, b = 56, c = 65
The first two triangles are clearly right-angled.
Is the third one ?
is 65^2 = 33^2 + 56^2 ??, YES
So make sketches of each one.
for a) I labeled by triangle ABC, where a = 30
angle A = 90, and angle B = 30°
sin35° = 30/b
bsin35 = 30
b = 30/sin35 = appr 52.30
tan35 = 30/c
c = 30/tan35 = appr 42.84
(after I found b, I could have used Pythagoras to find c
c^2 + 30^2 = 52.3^2
c^2 = 1835.29
c = 42.84, confirming my answer above )
So the others the same way
for the last one, obviously the 90° angle is opposite the largest side.
use tanØ to find one of the 2 missing angles, the third one is then obvious.
To find the remaining side lengths and angle measurements in the given triangles, we can use trigonometric ratios such as sine, cosine, and tangent. Let's solve each of the given scenarios one by one:
(a) α = 35°, β = 90°, a = 30:
First, let's determine the remaining angle, γ, by using the fact that the sum of the angles in a triangle is always 180°. Thus, γ = 180° - α - β = 180° - 35° - 90° = 55°.
Now we can use the Sine, Cosine, and Tangent ratios to find the missing side lengths:
1. To find side b, we can use the Sine ratio: sin α = b / a
sin 35° = b / 30
b = 30 * sin 35° ≈ 17.17 (rounded to 2 decimal places)
2. To find side c, we can use the Cosine ratio: cos β = c / a
cos 90° = c / 30
c = 30 * cos 90° = 0 (since cos 90° = 0)
3. To find side d, we can use the Sine ratio: sin γ = d / a
sin 55° = d / 30
d = 30 * sin 55° ≈ 24.11 (rounded to 2 decimal places)
The remaining side lengths are approximately b ≈ 17.17 and d ≈ 24.11.
The angle measurement γ is approximately 55°.
(b) α = 90°, b = 7, c = 6:
In this case, we already know that one angle is 90°, so it's a right triangle.
1. To find the missing angle β, we can use the Sine ratio: sin β = b / c
sin β = 7 / 6
β = sin^(-1)(7 / 6) ≈ 62.45° (rounded to 2 decimal places)
2. To find the missing side a, we can use the Pythagorean theorem: a^2 = c^2 - b^2
a^2 = 6^2 - 7^2
a^2 = 36 - 49
a^2 = -13 (impossible solution, since side lengths cannot be negative)
Therefore, in this scenario, angle β is approximately β ≈ 62.45°, and there is no possible value for the missing side length a.
(c) a = 33, b = 56, c = 65:
In this scenario, we know all three side lengths of the triangle.
1. To find angle α, we can use the Cosine ratio: cos α = (b^2 + c^2 - a^2) / (2 * b * c)
cos α = (56^2 + 65^2 - 33^2) / (2 * 56 * 65)
α = cos^(-1)((56^2 + 65^2 - 33^2) / (2 * 56 * 65)) ≈ 19.42° (rounded to 2 decimal places)
2. To find angle β, we can use the Cosine ratio: cos β = (a^2 + c^2 - b^2) / (2 * a * c)
cos β = (33^2 + 65^2 - 56^2) / (2 * 33 * 65)
β = cos^(-1)((33^2 + 65^2 - 56^2) / (2 * 33 * 65)) ≈ 30.86° (rounded to 2 decimal places)
3. To find angle γ, we can use the Cosine ratio: cos γ = (a^2 + b^2 - c^2) / (2 * a * b)
cos γ = (33^2 + 56^2 - 65^2) / (2 * 33 * 56)
γ = cos^(-1)((33^2 + 56^2 - 65^2) / (2 * 33 * 56)) ≈ 129.72° (rounded to 2 decimal places)
Therefore, in this scenario, angle α is approximately α ≈ 19.42°, angle β is approximately β ≈ 30.86°, and angle γ is approximately γ ≈ 129.72°.