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Calculate Angle x in a Right-Angled Triangle Using Trigonometry
Mathematics
Grade 9 (Junior High School)
Question Content
Shown below is a right-angled triangle with one leg of 3cm, hypotenuse of 8cm, and angle x adjacent to the 3cm leg. Use trigonometry to work out the size of angle x.
Correct Answer
Approximately 67.98° (or 68° rounded to the nearest whole number)
Detailed Solution Steps
1
Step 1: Identify the sides relative to angle x. The side adjacent to angle x is 3cm, and the hypotenuse of the right-angled triangle is 8cm.
2
Step 2: Select the appropriate trigonometric ratio. Since we have the adjacent side and hypotenuse, we use the cosine function: cos(x) = adjacent / hypotenuse.
3
Step 3: Substitute the known values into the formula: cos(x) = 3/8 = 0.375.
4
Step 4: Calculate angle x by finding the inverse cosine (arccos) of 0.375: x = arccos(0.375).
5
Step 5: Use a calculator in degree mode to compute the value: x ≈ 67.98°, which can be rounded to 68° for simplicity.
Knowledge Points Involved
1
Trigonometric Ratios (Cosine) in Right-Angled Triangles
For an acute angle in a right-angled triangle, cosine of the angle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse, written as cos(θ) = adjacent / hypotenuse. It is used to find unknown angles or side lengths when the adjacent side and hypotenuse are known.
2
Inverse Trigonometric Functions (Arccosine)
The arccosine function (arccos or cos⁻¹) is the inverse of the cosine function. It is used to find the measure of an acute angle in a right-angled triangle when the ratio of the adjacent side to the hypotenuse is known. If cos(θ) = k, then θ = arccos(k).
3
Calculator Usage for Trigonometric Calculations
When calculating inverse trigonometric values, ensure the calculator is set to the correct mode (degree mode for angle measurements in degrees). Input the trigonometric ratio and use the inverse function to get the corresponding angle measure.
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