Question

In a certain triangle, the measures of \(\angle A\) and \(\angle B\) are \((6 k-8)^{\circ}\) and \((7 k-\) 45) \(^{\circ},\) respectively. If \(\frac{\sin \angle A}{\cos \angle B}=1\), what is the value of \(k ?\)

Short answer

The value of k is 11.

Step-by-step solution

Understand the given relationship

The problem states that \( \frac{\sin \angle A}{\cos \angle B} = 1 \). Use the identity \( \sin x = \cos(90^{\circ} - x) \) to rewrite the equation.

Apply the trigonometric identity

Using \( \sin x = \cos(90^{\circ} - x) \), we have \( \frac{\sin \angle A}{\cos \angle B} = \frac{\cos(90^{\circ} - \angle A)}{\cos \angle B} \). Since the ratio is 1, it implies that \( 90^\circ - \angle A = \angle B \).

Set up the equation with given angles

Given \( \angle A = (6k - 8)^\circ \) and \( \angle B = (7k - 45)^\circ \), set up the equation \( 90^\circ - (6k - 8) = 7k - 45 \).

Simplify the equation

Simplify the equation to remove parentheses: \( 90 - 6k + 8 = 7k - 45 \). Combine like terms to get \( 98 - 6k = 7k - 45 \).

Solve for k

Add \( 6k \) and \( 45 \) to both sides: \( 98 + 45 = 7k + 6k \), resulting in \( 143 = 13k \). Then divide both sides by 13 to find \( k \): \( k = \frac{143}{13} = 11 \).

Key concepts

Understanding Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles. It's essential in various fields such as physics, engineering, and even art. One key concept in trigonometry is the trigonometric identities, which are equations involving trigonometric functions that hold true for all values of the variables involved. For instance, the identity \( \sin x = \cos(90^{\circ} - x) \) tells us that the sine of an angle is equal to the cosine of its complement. These identities are very useful in solving problems related to angles and triangles.

Triangle Properties

Triangles are three-sided polygons and have several important properties. One crucial property is that the sum of the interior angles of a triangle is always 180 degrees. Each triangle has three angles and three sides, with different names such as:

• Right Triangle: One angle is 90 degrees.
• Equilateral Triangle: All sides and angles are equal.
• Isosceles Triangle: Two sides and two angles are equal.
• Scalene Triangle: All sides and angles are different.

In our problem, we deal with general properties to find relationships between angles.

Angle Measures

In geometry, measuring angles is fundamental. Angles are often measured in degrees, where a full circle is 360 degrees. When solving problems involving angle measures, we use various techniques such as complementary and supplementary angles:

• Complementary angles add up to 90 degrees.
• Supplementary angles add up to 180 degrees.

For instance, using the identity \( \frac{\sin \angle A}{\cos \angle B} = 1\), we deduce that \( \angle A \) and \( \angle B \) are complementary angles in our problem.

Solving Algebraic Equations

Algebra is used everywhere in math to solve equations and find unknown values. To solve the given equation \( 90 - (6k - 8) = 7k - 45 \), we use algebraic manipulation:

• First we simplify and remove parentheses: \( 90 - 6k + 8 = 7k - 45 \).
• Next, combine like terms to form a simpler equation: \[98 - 6k = 7k - 45\].
• Then, isolate the variable \( k \) by moving terms around: \[98 + 45 = 7k + 6k\].
• Finally, solve for \( k \) by dividing both sides by \( 13 \): \[143 = 13k\] leading to \[ k = \frac{143}{13} = 11\].

This process helps in breaking down complex problems into manageable steps, ensuring we understand and find the correct solution efficiently.