➤ Activity 4: What Makes You Special? 1. Given the angles of the triangles below, find the values of the six trigonometric ratios. Then answer the questions that follow. Let a be the leg of a 45°-45°-90° Triangle. a√₂ 45° sin 45° = cos 45º = tan 45º = B 45° sec 45° = csc 45° = cot 45° = Let a be the shorter leg of a 30° -60°-90° Triangle. 30° sin 30° = cos 30º = tan 30º = 2a a√3 60° sec 30° = csc 30° = cot 30º =
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Answer:
In the given triangles, we can calculate the values of the trigonometric ratios using the known angles and side lengths.
For the 45°-45°-90° triangle:
- Given: one leg is "a"
- The other leg is also "a" (since it's an isosceles triangle)
- Hypotenuse = a√2 (since the sides are in the ratio 1:1:√2)
The trigonometric ratios for the 45° angle are:
- sin 45° = opposite/hypotenuse = a/a√2 = 1/√2 = √2/2
- cos 45° = adjacent/hypotenuse = a/a√2 = 1/√2 = √2/2
- tan 45° = opposite/adjacent = a/a = 1/1 = 1
- sec 45° = 1/cos 45° = 1/(√2/2) = 2/√2 = √2
- csc 45° = 1/sin 45° = 1/(√2/2) = 2/√2 = √2
- cot 45° = 1/tan 45° = 1/1 = 1
For the 30°-60°-90° triangle:
- Given: shorter leg = "a"
- Longer leg = 2a (since the sides are in the ratio 1:2:√3)
- Hypotenuse = 2a√3 (since the sides are in the ratio 1:2:√3)
The trigonometric ratios for the 30° angle are:
- sin 30° = opposite/hypotenuse = a/(2a√3) = 1/(2√3) = √3/6
- cos 30° = adjacent/hypotenuse = 2a/(2a√3) = √3/√3 = 1/2
- tan 30° = opposite/adjacent = a/(2a) = 1/2
- sec 30° = 1/cos 30° = 1/(1/2) = 2
- csc 30° = 1/sin 30° = 1/(√3/6) = 2√3/3
- cot 30° = 1/tan 30° = 1/(1/2) = 2
So, the values of the trigonometric ratios are:
For the 45°-45°-90° triangle:
sin 45° = cos 45° = √2/2
tan 45° = 1
sec 45° = csc 45° = √2
cot 45° = 1
For the 30°-60°-90° triangle:
sin 30° = √3/6
cos 30° = 1/2
tan 30° = 1/2
sec 30° = 2
csc 30° = 2√3/3
cot 30° = 2