Question

Evaluating inverse trigonometric functions Without using a calculator, evaluate or simplify the following expressions. $$\csc ^{-1}(\sec 2)$$

Short answer

Question: Evaluate the expression \(\csc ^{-1}(\sec 2)\) without using a calculator. Answer: \(88^{\circ}\)

Step-by-step solution

Understand the function notation

First, let's understand the notation for inverse trigonometric functions. The notation \(\csc ^{-1}(\sec 2)\) is another way of writing \(arc \csc (\sec 2)\). We want to find the angle whose cosecant is equal to the secant of 2.

Define the cosecant

Cosecant is the reciprocal function of sine. We can write the given function as: $$\csc ^{-1}(\sec 2) = arc \csc \left(\frac{1}{\sin (\theta)}\right) = \theta$$ Where we need to find the angle \(\theta\).

Use the reciprocal identity for secant

Secant is the reciprocal function of cosine. So, we can write \(\sec 2\) as: $$\sec 2 = \frac{1}{\cos(2)}$$ Now, we substitute this into the expression to get: $$arc \csc \left(\frac{1}{\sin (\theta)}\right) = arc \csc \left( \frac{1}{\cos(2)} \right) = \theta$$

Rewrite the expression in terms of sine and cosine

Now, we notice that the expression for arc cosecant is the reciprocal of sine. Therefore, we can rewrite the expression as follows: $$ \frac{1}{\sin(\theta)} = \frac{1}{\cos(2)}$$

Take the reciprocal of both sides and simplify

Taking the reciprocal on both sides of the equation, we have: $$\sin(\theta) = \cos(2)$$

Identify the angle "theta" which satisfies the relation

From the properties of a right-angled triangle, we know that sine and cosine are complementary functions i.e.,: $$\sin(\theta) = \cos(90^{\circ} - \theta)$$ Comparing with our expression, we have: $$\sin(\theta) = \cos(2)$$ Thus, the angle \(\theta\) satisfies the relation: $$\theta = 90^{\circ} - 2$$ So, the given expression evaluates to: $$\csc ^{-1}(\sec 2) = \theta = 90^{\circ} - 2 = 88^{\circ}$$

Key concepts

Cosecant

Cosecant, abbreviated as \(csc\), is one of the basic trigonometric functions. It is defined as the reciprocal of the sine function in trigonometry.
The term 'cosecant' originates from the Latin "cosecadent", meaning "opposite the arc."
Mathematically, if \(\sin(\theta)\) is the sine of angle \(\theta\), then the cosecant is given by:

• \(csc(\theta) = \frac{1}{\sin(\theta)}\)

This function is particularly useful when analyzing relationships in right-angle triangles and solving equations when the sine value is known or determined.
It is important to note that \(csc(\theta)\) is undefined for angles where \(\sin(\theta) = 0\), as division by zero is not possible in mathematics.
This means that the cosecant function is undefined at integer multiples of \(180^\circ\) or \(\pi\) radians.

Secant

The secant function, denoted as \(sec\), is another reciprocal trigonometric function. It acts as the reciprocal of the cosine function.
This makes \(sec(\theta)\) defined as:

• \(sec(\theta) = \frac{1}{\cos(\theta)}\)

Secant plays a crucial role when dealing with trigonometric identities and transformations.
It's especially handy when you need to explore or highlight relationships and properties involving the cosine side in a right triangle.
Like the cosecant function, secant is undefined when \(\cos(\theta) = 0\). Consequently, the secant function will be undefined for angles \(\theta = (2n+1)\cdot90^\circ\) or \((2n+1)\cdot\frac{\pi}{2}\) radians, where \(n\) is an integer.

Trigonometric Identities

Trigonometric identities are equations comprising trigonometric functions that are true for every possible value of the occurring variables.
They are vital tools for simplifying complex trigonometric expressions and solving trigonometric equations.

• Reciprocal Identities: These identities help to express trigonometric functions in terms of their reciprocals. For example, \(sec(\theta) = \frac{1}{\cos(\theta)}\) and \(csc(\theta) = \frac{1}{\sin(\theta)}\).
• Pythagorean Identities: These arise from the Pythagorean theorem and include equations like \(sin^2(\theta) + cos^2(\theta) = 1\).

When solving problems such as \(\csc^{-1}(\sec(2))\), these identities allow us to simplify and interpret expressions more effectively.
They serve as useful shortcuts and provide deeper insights into the geometric nature of these functions.
Being proficient with these identities can greatly facilitate the process of evaluating and simplifying trigonometric expressions, as seen in our exercise.

Right-Angle Trigonometry

Right-angle trigonometry deals with relationships between the angles and sides of right triangles.
This area of mathematics provides fundamental insights that are utilized in various applications, from geometry to engineering.
In the context of right triangles, the functions sine, cosine, and tangent define specific ratios of a triangle's sides:

• Sine (\(sin\)): Ratio of the length of the opposite side to the hypotenuse.
• Cosine (\(cos\)): Ratio of the length of the adjacent side to the hypotenuse.
• Tangent (\(tan\)): Ratio of the length of the opposite side to the adjacent side.

The reciprocal functions — cosecant, secant, and cotangent — provide additional relationships and can be useful in solving equations where these ratios occur.
Understanding right-angle trigonometry is crucial when working with inverse trigonometric functions, as they are often used to find angles in problems related to constructions or spatial reasoning.
With these foundational concepts, evaluating expressions involving inverses like \(\csc^{-1}(\sec(2))\) becomes more intuitive.