Question

(i) In a \(\triangle \mathrm{ABC}\), if \(\mathrm{E}\) is the point on the base \(\mathrm{CA}\) such that \(\frac{\mathrm{CE}}{\mathrm{EA}}=\frac{3}{5}\) and \(\angle \mathrm{AEB}=\theta\), show that \(8 \cot \theta=5 \cot \mathrm{C}-3 \cot \mathrm{A}\) (ii) Find the value of \(\cot \theta\) if it is given that \(\sin C=\frac{5}{13}\) and \(\sin A=\frac{3}{5}\).

Short answer

Question: Show that \(8cotθ = 5cotC - 3cotA\) for a triangle ABC with point E on base CA such that \(CE/EA = 3/5\), \(\angle AEB = θ\), \(\angle ACB = C\), and \(\angle ABC = A\). Also, find the value of cotθ if sin(C) = 5/13 and sin(A) = 3/5. Answer: We showed that \(8cotθ = 5cotC - 3cotA\) by using the Law of Sines and trigonometric identities, and relating the cotangent function to the side lengths of the triangle. Then, we found the cotangent of angles A and C given their sine values. By plugging these values into the proven equality, we found that the value of cotθ is 1.

Step-by-step solution

Draw the triangle and label the sides and angles

Draw a triangle ABC and mark the point E on the base CA. Label angle AEB as θ, angle ACB as C, and angle ABC as A. Let \(CE = 3x\) and \(EA = 5x\). Then, since \(\angle AEB\) is opposite the side \(EA\), we can denote the sides of the triangle by \(AB=a\), \(BC=b\), and \(AC=c\).

Apply Law of Sines on Triangle ABE

Applying the Law of Sines on triangle ABE, we have: \(\frac{a}{\sin \theta} = \frac{5x}{\sin A}\). Rearrange the equation to get: \(a \sin A = 5x \sin \theta\). (1)

Apply Law of Sines on Triangle CBE

Applying the Law of Sines on triangle CBE, we have: \(\frac{b}{\sin \theta} = \frac{3x}{\sin C}\). Rearrange the equation to get: \(b \sin C = 3x \sin \theta\). (2)

Divide equations (1) and (2)

Now, we will divide the equation (2) by equation (1) to eliminate \(x \sin \theta\) term: \(\frac{b \sin C}{a \sin A} = \frac{3x \sin \theta}{5x \sin \theta}\). Cancelling out \(x\sin\theta\) terms gives: \(\frac{b \sin C}{a \sin A} = \frac{3}{5}\).

Find cotangent of the angles

Express the cotangent of the angles A and C in terms of sin and cos: \(\cot A = \frac{\cos A}{\sin A}\) and \(\cot C = \frac{\cos C}{\sin C}\).

Relate cotangent to triangle side lengths

Now, we will relate the cotangent of the angles A and C to the side lengths: \(5 \cot C = 5 \cdot \frac{\cos C}{\sin C} = \frac{a}{b}\) and \(3 \cot A = 3 \cdot \frac{\cos A}{\sin A} = \frac{b}{a}\).

Show the required equality

From the previous step, we can now prove the required equality: \(8\cot \theta = 5\cot C - 3\cot A = 5 \cdot \frac{a}{b} - 3 \cdot \frac{b}{a} = \frac{5a^2 - 3b^2}{ab}\). Therefore, we have proved that \(8 \cot \theta = 5 \cot C - 3 \cot A\).

Find the value of cotθ

We are given \(\sin C = \frac{5}{13}\) and \(\sin A = \frac{3}{5}\), so we can find the corresponding cosines using Pythagorean identities: \(\cos C = \sqrt{1 - \sin^2 C} = \frac{12}{13}\) and \(\cos A = \sqrt{1 - \sin^2 A} = \frac{4}{5}\). Now we can find the cotangent values: \(\cot C = \frac{\cos C}{\sin C} = \frac{12}{5}\) and \(\cot A = \frac{\cos A}{\sin A} = \frac{4}{3}\). Substitute the cotangent values into the proven equality: \(8 \cot \theta = 5 \cdot \frac{12}{5} - 3 \cdot \frac{4}{3}\). Solve for cotθ: \(\cot \theta = \frac{12 - 4}{8} = 1\). Hence, the value of cotθ is 1.

Key concepts

Cotangent

When studying trigonometry, you'll come across various functions that are essential for understanding the relationships between angles and sides in triangles. One such function is the cotangent. The cotangent of an angle is defined as the ratio of the adjacent side to the opposite side in a right-angled triangle. In formula terms, cotangent is written as \( \text{cot} \) and given by the equation \( \text{cot} \theta = \frac{\text{adjacent}}{\text{opposite}} \). Another way to express the cotangent is the reciprocal of the tangent function: \( \text{cot} \theta = \frac{1}{\text{tan} \theta} \). In the context of the problem, we used the cotangent to express the relationship between angles and sides within a triangle that does not necessarily have a right angle.

It's useful to remember that cotangent is also the ratio of the cosine function to the sine function: \( \text{cot} \theta = \frac{\text{cos} \theta}{\text{sin} \theta} \). This identity is particularly helpful in solving problems involving non-right triangles, such as in the given exercise where we used the Law of Sines to find the side lengths in terms of sine and cosine and eventually the cotangents of the angles.

Trigonometric Identities

Trigonometric identities are equations that hold true for all possible values of the variable angles. These identities are fundamental in simplifying complex trigonometric expressions and solving equations. The most foundational of these are the Pythagorean identities, which relate the squares of the sine and cosine of an angle. One of these identities is \( \text{sin}^2 \theta + \text{cos}^2 \theta = 1 \), which allows us to find one function value when the other is known.

For instance, in our exercise, given the value of \( \text{sin} C \) and \( \text{sin} A \), we used the respective identities to find the cosine values for these angles. Understanding how to manipulate and apply these identities is pivotal in solving trigonometry problems, as demonstrated by finding \( \text{cot} \theta \) in the step by step solution.

Triangle Properties

Triangles are three-sided polygons with inherent properties and theories that help us explore geometric concepts. Key properties include the fact that the sum of the interior angles of a triangle always add up to \( 180^\text{o} \), and the Law of Sines and Law of Cosines which are central in solving triangles; meaning, finding the unknown sides and angles of a triangle.

These laws are especially useful for non-right-angled triangles. For example, the Law of Sines, stated as \( \frac{\text{a}}{\text{sin} A} = \frac{\text{b}}{\text{sin} B}= \frac{\text{c}}{\text{sin} C} \), where \( \text{a} \) ,\( \text{b} \) , and \( \text{c} \) are sides opposite angles \( A \) ,\( B \) , and \( C \) of the triangle, was used in our exercise to relate the side lengths to the angles at E. Understanding these properties allows students to dissect complex geometric problems into manageable calculations.

Sine and Cosine

The sine and cosine functions are fundamental trigonometric functions. The sine of an angle in a right triangle is the ratio of the length of the opposite side to the length of the hypotenuse, while the cosine is the ratio of the length of the adjacent side to the hypotenuse. These functions are not just for right triangles; they're extended to all angles, positive or negative, and can even represent angles greater than \( 360^\text{o} \), usually in a unit circle context.

In the context of our exercise, \( \text{sin} A \) and \( \text{sin} C \) are used not only to express the relationship between sides but also to find the cotangent values, since \( \text{cot} \theta \) is finding the ratio of cosine and sine. Whenever working with trigonometric equations, you'll see these functions being used to depict the ratios of sides of triangles, and it's crucial to have a firm grasp on converting between them, as they often provide the gateway to the solution.