A rectangle is 48 cm long and 14 cm wide. If the diagonal makes an angle θ with the longer side, then what is (sec θ + cosec θ) equal to?
Understanding the Rectangle and its Diagonal Angle
Let's break down this geometry and trigonometry problem. We have a rectangle with a given length and width. The diagonal of the rectangle forms a right-angled triangle with the sides. We are interested in the angle the diagonal makes with the longer side.
Given:
- Length of the rectangle (longer side) = 48 cm
- Width of the rectangle (shorter side) = 14 cm
- \(\theta\) is the angle the diagonal makes with the longer side.
When we consider the diagonal, the longer side, and the shorter side of the rectangle, they form a right-angled triangle. The angle \(\theta\) is one of the acute angles in this triangle.
In the right-angled triangle formed by the longer side, shorter side, and the diagonal:
- The side adjacent to angle \(\theta\) is the longer side, with length 48 cm.
- The side opposite to angle \(\theta\) is the shorter side, with length 14 cm.
- The hypotenuse is the diagonal of the rectangle.
Calculating the Length of the Diagonal
We can find the length of the diagonal using the Pythagorean theorem in the right-angled triangle:
\(\text{(Diagonal)}^2 = \text{(Longer Side)}^2 + \text{(Shorter Side)}^2\)
Let 'd' be the length of the diagonal.
\(d^2 = 48^2 + 14^2\)
Let's calculate the squares:
\(48^2 = 48 \times 48 = 2304\)
\(14^2 = 14 \times 14 = 196\)
Now, sum them up:
\(d^2 = 2304 + 196\)
\(d^2 = 2500\)
Take the square root to find the diagonal length:
\(d = \sqrt{2500}\)
\(d = 50\) cm
So, the length of the diagonal is 50 cm.
Finding Trigonometric Ratios: Cosine and Sine of \(\theta\)
Now that we have all three sides of the right-angled triangle (adjacent = 48 cm, opposite = 14 cm, hypotenuse = 50 cm) with respect to angle \(\theta\), we can find the basic trigonometric ratios.
Cosine of \(\theta\):
\(\cos \theta = \frac{\text{Adjacent Side}}{\text{Hypotenuse}}\)
\(\cos \theta = \frac{48}{50}\)
Simplify the fraction:
\(\cos \theta = \frac{24}{25}\)
Sine of \(\theta\):
\(\sin \theta = \frac{\text{Opposite Side}}{\text{Hypotenuse}}\)
\(\sin \theta = \frac{14}{50}\)
Simplify the fraction:
\(\sin \theta = \frac{7}{25}\)
Calculating Secant and Cosecant of \(\theta\)
We need to find \(\sec \theta\) and \(\csc \theta\). These are the reciprocals of \(\cos \theta\) and \(\sin \theta\), respectively.
Secant of \(\theta\):
\(\sec \theta = \frac{1}{\cos \theta}\)
\(\sec \theta = \frac{1}{\frac{24}{25}}\)
\(\sec \theta = \frac{25}{24}\)
Cosecant of \(\theta\):
\(\csc \theta = \frac{1}{\sin \theta}\)
\(\csc \theta = \frac{1}{\frac{7}{25}}\)
\(\csc \theta = \frac{25}{7}\)
Finding the Value of (\(\sec \theta + \csc \theta\))
Now we can calculate the sum of \(\sec \theta\) and \(\csc \theta\):
\(\sec \theta + \csc \theta = \frac{25}{24} + \frac{25}{7}\)
To add these fractions, we need a common denominator. The least common multiple of 24 and 7 is \(24 \times 7 = 168\).
Convert each fraction to have the denominator 168:
\(\frac{25}{24} = \frac{25 \times 7}{24 \times 7} = \frac{175}{168}\)
\(\frac{25}{7} = \frac{25 \times 24}{7 \times 24} = \frac{600}{168}\)
Now add the fractions:
\(\sec \theta + \csc \theta = \frac{175}{168} + \frac{600}{168}\)
\(\sec \theta + \csc \theta = \frac{175 + 600}{168}\)
\(\sec \theta + \csc \theta = \frac{775}{168}\)
This is the required value.
| Measurement | Value |
|---|---|
| Rectangle Length | 48 cm |
| Rectangle Width | 14 cm |
| Diagonal Length | 50 cm |
| \(\cos \theta\) | \(\frac{24}{25}\) |
| \(\sin \theta\) | \(\frac{7}{25}\) |
| \(\sec \theta\) | \(\frac{25}{24}\) |
| \(\csc \theta\) | \(\frac{25}{7}\) |
| \(\sec \theta + \csc \theta\) | \(\frac{775}{168}\) |
Revision Table: Key Trigonometric Definitions
| Ratio | Definition |
|---|---|
| \(\sin A\) (Sine) | \(\frac{\text{Opposite Side}}{\text{Hypotenuse}}\) |
| \(\cos A\) (Cosine) | \(\frac{\text{Adjacent Side}}{\text{Hypotenuse}}\) |
| \(\tan A\) (Tangent) | \(\frac{\text{Opposite Side}}{\text{Adjacent Side}}\) |
| \(\csc A\) (Cosecant) | \(\frac{1}{\sin A} = \frac{\text{Hypotenuse}}{\text{Opposite Side}}\) |
| \(\sec A\) (Secant) | \(\frac{1}{\cos A} = \frac{\text{Hypotenuse}}{\text{Adjacent Side}}\) |
| \(\cot A\) (Cotangent) | \(\frac{1}{\tan A} = \frac{\text{Adjacent Side}}{\text{Opposite Side}}\) |
Additional Information: Angles in a Rectangle
A rectangle has four interior angles, each measuring 90 degrees. The diagonal divides the rectangle into two congruent right-angled triangles. If the angle the diagonal makes with the longer side is \(\theta\), then the angle it makes with the shorter side is \(90^\circ - \theta\). The sum of the angles in any triangle is \(180^\circ\), and in the right triangle formed by the diagonal, the angles are \(90^\circ\), \(\theta\), and \(90^\circ - \theta\).
The trigonometric ratios for \(90^\circ - \theta\) are related to those of \(\theta\):
- \(\sin(90^\circ - \theta) = \cos \theta\)
- \(\cos(90^\circ - \theta) = \sin \theta\)
- \(\sec(90^\circ - \theta) = \csc \theta\)
- \(\csc(90^\circ - \theta) = \sec \theta\)
These complementary angle relationships are fundamental in trigonometry.
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