Question

Evaluate the expression without using a calculator. (Hint: Make a sketch of a right triangle, as illustrated in Example \(7 .)\) (a) \(\sin \left(\arctan \frac{3}{4}\right)\) (b) \(\sec \left(\arcsin \frac{4}{5}\right)\)

Short answer

The solutions are: (a) \(\sin \left(\arctan \frac{3}{4}\right) = \frac{3}{5}\) and (b) \(\sec \left(\arcsin \frac{4}{5}\right) = \frac{5}{3}\).

Step-by-step solution

Evaluate \(\sin \left(\arctan \frac{3}{4}\right)\)

Begin by sketching a right triangle for the angle whose tangent is \(\frac{3}{4}\). The tangent of an angle in a right triangle is equivalent to the ratio of the opposite side to the adjacent side. So, create a right triangle where the adjacent side is 4 (the denominator), and the opposite side is 3 (the numerator). By Pythagorean theorem, the hypotenuse of this triangle is \( \sqrt{3^2 + 4^2} = 5\). Then, because \(\sin\) is the ratio of the opposite side to the hypotenuse, the expression simplifies to be \(\sin \left(\arctan \frac{3}{4}\right) = \frac{3}{5}\).

Evaluate \(\sec \left(\arcsin \frac{4}{5}\right)\)

Now, sketch another right triangle for the angle whose sine is \(\frac{4}{5}\). The sine of an angle in a right triangle is the ratio of the opposite side to the hypotenuse. So, construct a right triangle where the hypotenuse is 5 (the denominator), and the opposite side is 4 (the numerator). By the Pythagorean theorem, the adjacent side of this triangle is \(\sqrt{5^2-4^2} = 3\). Then, we know that secant is the reciprocal of cosine, and cosine is the ratio of the adjacent side to the hypotenuse. The expression simplifies to be \(\sec \left(\arcsin \frac{4}{5}\right) = \frac{5}{3}\).

Key concepts

Pythagorean Theorem

The Pythagorean theorem is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be expressed as the equation:
\[ a^2 + b^2 = c^2 \]
where \( c \) represents the length of the hypotenuse, and \( a \) and \( b \) represent the lengths of the triangle's other two sides. Understanding this theorem is crucial when dealing with right triangle trigonometry as it allows us to find an unknown side length given the other two. In the context of trigonometric identities, the Pythagorean theorem helps us see the inherent relationship between sine, cosine, and hypotenuse in right-angled triangles, which is instrumental in simplifying trigonometric expressions.

Inverse Trigonometric Functions

Inverse trigonometric functions, also known as arc functions, are the inverse operations of the basic trigonometric functions. They are used to determine the angle that corresponds to a specific trigonometric ratio. The three main inverse trigonometric functions are arcsine \( (\arcsin) \), arccosine \( (\arccos) \), and arctangent \( (\arctan) \).
For instance, if the sine of an angle \( \theta \) is \( \frac{4}{5} \), written as \( \sin(\theta) = \frac{4}{5} \), then \( \theta \) can be expressed as \( \theta = \arcsin(\frac{4}{5}) \). These inverse functions are key components when solving trigonometric equations or evaluating expressions, allowing us to work backwards from a known ratio to find an angle.

Right Triangle Trigonometry

Right triangle trigonometry is the study of the relationships between the angles and sides of right triangles. The basic trigonometric ratios are sine (sin), cosine (cos), and tangent (tan), which are functions of an angle of the triangle. The sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse. Likewise, cosine is the ratio of the adjacent side to the hypotenuse, and tangent is the ratio of the opposite side to the adjacent side.
These relationships are essential for solving many problems in trigonometry and can be easily recalled using mnemonic devices, such as 'SOHCAHTOA', which stands for Sine equals Opposite over Hypotenuse, Cosine equals Adjacent over Hypotenuse, and Tangent equals Opposite over Adjacent. The inverse trigonometric functions serve as the bridge between trigonometric ratios and the angles themselves.

Sine and Cosine Ratios

Sine and cosine ratios are foundational to trigonometry. The sine ratio (sin) compares the length of the side of the triangle opposite the angle to the length of the hypotenuse. The cosine ratio (cos), on the other hand, compares the length of the adjacent side of the angle to the hypotenuse. These ratios are used to define the fundamental trigonometric functions for the angle in the context of a unit circle as well.
Using the right triangle approach, if we have a triangle with an angle \( \theta \), then the sine of angle \( \theta \) is defined as \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \), and the cosine of angle \( \theta \) is defined as \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \). These ratios are not only theoretical constructs; they have practical applications in various fields such as physics, engineering, and geometry.