Question

The angle \(\theta\) between the two equal sides of an isosceles triangle measures \(0.53 \pm 0.005\) radian. The two equal sides are exactly 151 centimeters long. Calculate the length of the third side with an estimate for the absolute error and the relative error.

Short answer

The third side is approximately 79.61 cm with an absolute error of about 1.21 cm and a relative error of 1.52%.

Step-by-step solution

Identify the Known Values

We know the angle \( \theta \) between the two equal sides is \( 0.53 \pm 0.005 \) radians, and the two equal sides of the isosceles triangle are \( 151 \) cm each. We need to find the length of the third side and its errors.

Use the Law of Cosines

In an isosceles triangle, the length of the third side \( c \) can be calculated using the law of cosines: \[ c = \sqrt{2a^2(1 - \cos \theta)} \] where \( a = 151 \) cm is the length of the equal sides. Plug in the values: \[ c = \sqrt{2 \times 151^2 \times (1 - \cos 0.53)} \]

Calculate the Cosine of the Angle

Find the cosine of \( \theta = 0.53 \) radians: \( \cos(0.53) \approx 0.861 \). Substitute this into the formula to find \[ c = \sqrt{2 \times 151^2 \times (1 - 0.861)} \].

Simplify to Find the Third Side

Simplify the expression: \[ 1 - 0.861 = 0.139 \] and \[ c = \sqrt{2 \times 151^2 \times 0.139} \]. Calculate each part: \[ 151^2 = 22801 \] and \[ 2 \times 22801 \times 0.139 \approx 6333.98 \]. The square root gives \[ c \approx 79.61 \text{ cm} \].

Calculate Absolute Error

The absolute error in \( c \) is calculated by differentiating the formula \( c = \sqrt{2a^2(1 - \cos \theta)} \) with respect to \( \theta \). This gives: \[ \frac{dc}{d\theta} = \frac{a^2 \sin \theta}{c} \]. Substitute the values: \( \sin(0.53) \approx 0.848 \), and approximate the derivative: \[ \Delta c \approx \frac{151^2 \times 0.848}{79.61} \times 0.005 \approx 1.21 \text{ cm} \].

Calculate Relative Error

The relative error is calculated by dividing the absolute error by the calculated length of the third side: \[ \text{Relative error} = \frac{1.21}{79.61} \approx 0.0152 \]. Expressed in percentage, the relative error is approximately \( 1.52\% \).

Key concepts

Isosceles Triangle

An isosceles triangle is a special kind of triangle where two of its sides are equal in length. These two sides are called the 'legs', and the angles opposite these sides are also equal. This unique property simplifies many calculations, including those involving the application of trigonometry.

In an isosceles triangle, the unequal side is commonly referred to as the 'base'. In problems where we need to find the length of this base, the law of cosines becomes extremely helpful. This is because the angle included between the two equal sides (legs) can be used explicitly to find out the measure of the base, making the computation straightforward and reliant on familiar trigonometric functions.

Absolute Error

Absolute error is a measure of how much the calculated or measured value deviates from the true or exact value. It signifies the magnitude of the discrepancy without regard to its direction, so it’s always a positive value.

In calculations involving angles and sides of triangles, small variations in measurement precision can lead to an absolute error when computing, for example, the side lengths. This measure is crucial for understanding the accuracy of a calculation. To find it in our exercise, we considered the differential of the length of the third side concerning the angle, and it turned out to be approximately 1.21 cm.

Understanding absolute error helps in assessing and improving the reliability of your calculated results by making allowance for the potential discrepancies.

Relative Error

Relative error is used to measure the size of the absolute error relative to the total size of the measurement. It provides a way to gauge precision by comparing the absolute error with the magnitude of the measured or calculated value.

To compute relative error, one would typically divide the absolute error by the measured or calculated value. It is often expressed as a percentage to give a clear sense of how large the error is compared to the actual measurement.

In the context of the triangle exercise, the relative error of the third side was found to be approximately 1.52%. This means that the possible error in the length of the third side is about 1.52% of its actual size, which gives you an idea of the reliability of this measurement.

Trigonometry

Trigonometry is the branch of mathematics that studies relationships involving lengths and angles of triangles. It is particularly useful in solving problems related to right and non-right triangles, including isosceles triangles.

Key functions in trigonometry include sine, cosine, and tangent, which relate the angles of a triangle to the lengths of its sides. The law of cosines, which we applied in this exercise, is a crucial tool in trigonometry.

By expressing the third side of an isosceles triangle in terms of known side lengths and the angle between them, trigonometry allows you to convert angular information into linear information. This makes it possible to calculate unknown sides when the other sides and an angle are known, as shown when using cosine in our problem to find the length of the third side of the triangle.