Question

Find the exact value of each expression. Do not use a calculator. $$ 1+\tan ^{2} 30^{\circ}-\csc ^{2} 45^{\circ} $$

Short answer

\( -\frac{2}{3} \)

Step-by-step solution

- Recall key trigonometric values

First, identify and recall the trigonometric values for the given angles: For \( 30^{\circ} \): \( \tan(30^{\circ}) = \frac{1}{\sqrt{3}} \). For \( 45^{\circ} \): \( \csc(45^{\circ}) = \frac{\sqrt{2}}{1} = \sqrt{2} \).

- Compute \( \tan^2 30^{\circ} \)

Square the tangent value: \( \tan^2 30^{\circ} = \left( \frac{1}{\sqrt{3}} \right)^2 = \frac{1}{3} \).

- Compute \( \csc^2 45^{\circ} \)

Square the cosecant value: \( \csc^2 45^{\circ} = ( \sqrt{2} )^2 = 2 \).

- Substitute the values into the expression

Replace the trigonometric values in the original expression: \( 1 + \tan^2 30^{\circ} - \csc^2 45^{\circ} \) becomes \( 1 + \frac{1}{3} - 2 \).

- Simplify the expression step-by-step

Combine the values: First, find a common denominator to combine 1 and \( \frac{1}{3}\): \( 1 = \frac{3}{3} \), thus \( \frac{3}{3} + \frac{1}{3} = \frac{4}{3} \). Now subtract 2: \( \frac{4}{3} - 2 = \frac{4}{3} - \frac{6}{3} = -\frac{2}{3} \).

Key concepts

Understanding the Tangent of 30 Degrees

To grasp what the tangent of 30 degrees means, let's recall what the tangent function represents. Tangent, denoted as \( \tan( \theta ) \), is a trigonometric ratio comparing the opposite side to the adjacent side of a right triangle at a given angle \( \theta \). For 30 degrees, the right triangle can often be considered as part of a special 30-60-90 triangle.
The 30-60-90 triangle has side lengths that follow the ratio 1: \( \frac{\text{1}}{\text{2}} \) : \( \frac{\text{1}}{\text{\text{ √ 3 }}} \). In this triangle, if the shortest side (opposite the 30-degree angle) is 1, then:

• The hypotenuse is 2.
• The side adjacent to the 30-degree angle is \( \frac{√3}{3} \).

Therefore, \( \tan (30^{\text{\text{ o }}}) = \frac{1} {\text{\text{ √ 3 }}} \)
Squaring this, we get: \( \tan ^ {2}(30^{\text{\text{ o }}}) = (\frac{1}{\text{\text{ √3 }}})^2 = \frac{1}{3} \)
Knowing this value is crucial for understanding related trigonometric operations and expressions.

Exploring the Cosecant of 45 Degrees

Cosecant (\( \text{csc} \)) is the reciprocal of the sine function. It is represented as \( \text{csc}( \theta ) = \frac{1}{ \text{sin} ( \theta )} \). When dealing with a 45-degree angle, we look at the special 45-45-90 triangle.
In a 45-45-90 triangle, both legs are equal and the hypotenuse is \( \text{√ 2 }\) times a leg. Hence:

• \( \text{sin}(45^{\text{\text{ o }}}) = \frac{\text{\text{Opposite Side}}}{\text{\text{Hypotenuse}}} = \frac{1}{ \text{√ 2 }} = \frac{ \text{ √ 2 }}{2} \).

Taking the reciprocal, we get:
\( \text{csc}(45^{\text{\text{ o}}}) = \frac{1}{ \text{sin}(45^{\text{\text{ o }}})} = \frac{1}{ \frac{\text{ √ 2 }}{ 2}} = \text{ √ 2} \)
Squaring this, we have: \( \text{csc}^{ \text{ 2 }}(45^{ \text{ o }}) = (\text{ √ 2 })^{ \text{\text{ 2 }}} = 2 \)
This value plays a fundamental role in simplifying complex trigonometric expressions.

Understanding Exact Trigonometric Values

Exact trigonometric values are those that we can determine without a calculator, often derived from standard angles like 30, 45, and 60 degrees. These values are memorized or derived from specific triangles: the 30-60-90 and 45-45-90 triangles.
For these standard angles, we know the following key values:

• \( \text{\text{sin}( 30^{\text{\text{ o }}} ) = \frac{1 }{ 2 } \)
• \( \text{cos}( 30^{\text{\text{ o }}} ) = \text{\text{ √ 3 }}/ 2 \)
• \( \tan( 30^{\text{\text{ o }}} ) = \frac{1}{\text{\text{ √ 3}}} \)
• \( \text{sin}( 45^{\text{\text{ o }}} ) = \frac{\text{\text{ √ 2 }}}{ 2 } \)
• \( \text{cos}( 45^{\text{\text{ o }}} ) = \frac{\text{\text{ √ 2 }}}{ 2 } \)
• \( \text{\text{\tan(45^{\text{\text{ o }}} )}} = 1 \)

These values allow us to solve many trigonometric problems without relying on calculators. They form the backbone of deeper trigonometric understanding and manipulation.
For example, they enable us to evaluate complex expressions like the one in this exercise: \( 1+ \tan^{ 2 } ( 30^{ \text{ o } }) - \text{csc}^{ 2}( 45^{\text{ o }}) = 1 + \frac{ 1}{3} - 2 = -\frac{2}{3} \) Utilizing these precise values improves accuracy and helps build a strong foundation in trigonometry.