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Calculate the value of angle y in a right-angled triangle using trigonometry
Mathematics
Grade 10 of Junior High School
Question Content
Work out the value of y in the right-angled triangle with adjacent side to angle y being 8 cm and opposite side to angle y being 9 cm. Give your answer correct to 1 decimal place where appropriate.
Correct Answer
48.4°
Detailed Solution Steps
1
Step 1: Identify the sides relative to angle y. The side opposite angle y is 9 cm, and the side adjacent to angle y is 8 cm.
2
Step 2: Select the appropriate trigonometric ratio. Since we have opposite and adjacent sides, we use the tangent function: tan(y) = opposite/adjacent.
3
Step 3: Substitute the known values into the formula: tan(y) = 9/8 = 1.125.
4
Step 4: Calculate y by taking the inverse tangent (arctangent) of 1.125: y = arctan(1.125).
5
Step 5: Use a calculator to compute the value, rounding to 1 decimal place: y ≈ 48.4°.
Knowledge Points Involved
1
Trigonometric Ratios (Tangent Function)
In a right-angled triangle, the tangent of an acute angle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle, written as tan(θ) = opposite/adjacent. It is used to relate angles and side lengths in right triangles.
2
Inverse Trigonometric Functions (Arctangent)
The arctangent function (arctan or tan⁻¹) is the inverse of the tangent function. It is used to find the measure of an acute angle in a right triangle when the ratio of the opposite and adjacent sides is known, i.e., if tan(θ) = x, then θ = arctan(x).
3
Rounding Decimal Places
Rounding to 1 decimal place means adjusting a number so that it has only one digit after the decimal point. If the second decimal digit is 5 or greater, the first decimal digit is increased by 1; otherwise, it stays the same. This is used to present numerical answers to a specified level of precision.
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