Determine each ratio as a decimal to four places, then find the angle to the nearest whole number A Ratio Ratio as a decimal Angle sin c 15 cm 8 cm COS C ĐI tan C С 17 cm B COS 8 sin B
Mathematics · High School · Tue Nov 03 2020
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To find the ratios as decimals and their corresponding angles, we need to perform some trigonometric calculations based on the provided measurements of a right-angled triangle ABC (assuming it is a right-angled triangle with angle C being the right angle):
1. **sin C** - We have opposite side = 8 cm and the hypotenuse = 15 cm. Therefore, sin C = opposite/hypotenuse = 8/15 ≈ 0.5333 (to four places).
To find the angle C, we'd use the inverse sine function (arcsin), but since C is a right angle in our assumption, this step wouldn't be necessary. However, if we are required to find the angle C that isn't a right angle (which is contradictory with initial assumptions), we would use: C = arcsin(0.5333) ≈ 32.3° (to the nearest whole number would be 32°).
2. **cos C** - This is the adjacent side over the hypotenuse. Since angle C is the right angle, cos C is actually 0, as cosine of 90 degrees is 0. If angle C is not 90 degrees and we have the adjacent side measurement, we would need that value.
3. **tan C** - Since angle C is the right angle in the triangle, tan C is undefined because tangent is the ratio of the opposite side to the adjacent side (opposite/adjacent), and the length of the adjacent side approaches 0 as the angle approaches 90 degrees.
If we're given an opposite side (8 cm) and an adjacent side (17 cm) instead, assuming the question is about angle B: 1. **cos B** - We are given the adjacent side, 17 cm, and the hypotenuse, which is missing. We calculate the hypotenuse using Pythagoras' theorem:
\( a^2 + b^2 = c^2 \) \( 8^2 + 17^2 = c^2 \) \( 64 + 289 = c^2 \) \( c^2 = 353 \) \( c ≈ 18.7883 cm \)
cos B = adjacent/hypotenuse = 17/18.7883 ≈ 0.9046 (to four places). To find B, we take the inverse cosine (arccos): B = arccos(0.9046) ≈ 25.3° (to the nearest whole number would be 25°).
2. **sin B** - Since we have the opposite side (8 cm) and the hypotenuse (18.7883 cm), sin B = opposite/hypotenuse = 8/18.7883 ≈ 0.4258 (to four places). To find angle B, we could also use the inverse sine function: Angle B = arcsin(0.4258) ≈ 25.3° (which matches our previous calculation, since sin and cos are complementary for the two non-right angles in a right-angled triangle).