Problem 66 - Algebra and Trigonometry

Question

Find the exact value of each expression. Do not use a calculator. $$ 1-\tan ^{2} 10^{\circ}+\csc ^{2} 80^{\circ} $$

Step-by-Step Solution

Answer

The exact value of the expression is 2

Step 1: Apply Angle Properties

Recognize that \(10^{\circ}\) and \(80^{\circ}\) are complementary angles, i.e. they sum to \(90^{\circ}\). Thus \(\tan(10^{\circ}) = \cot(80^{\circ})\) and \(\csc(80^{\circ}) = \sec(10^{\circ})\). Thus, the expression can be rewritten as \(1-\tan ^{2} 10^{\circ}+\sec ^{2} 10^{\circ}\)

Step 2: Apply Pythagorean Identity

The trigonometric identity to be used is \(\sec ^{2} x = 1+\tan ^{2} x\). Thus, the expression is equivalent to \((1-\tan ^{2} 10^{\circ}) + (1+\tan ^{2} 10^{\circ})\)

Step 3: Perform the Operation

Perform the addition operation, which simplifies things, as \(1-\tan ^{2} 10^{\circ} + 1+\tan ^{2} 10^{\circ} = 1 + 1 = 2\)

Key Concepts

Complementary Angles

Complementary angles are pairs of angles whose measures add up to 90 degrees. In trigonometry, complementary angles have interesting relationships, especially for the sine, cosine, and tangent functions.
If two angles are complementary, like in our example with angles 10° and 80°, a helpful rule is that the sine of one angle is the cosine of the other. Similarly, the tangent of one angle is the cotangent of the other.

• For the given exercise, recognizing that 10° and 80° are complementary allows us to rewrite trigonometric functions. For example, \( an(10^{\circ}) = \cot(80^{\circ})\).


• This relationship simplifies the expressions and calculations inside trigonometric expressions.

Understanding this concept helps in rewriting and solving complex trigonometric expressions easier.

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry. It is related to the Pythagorean Theorem and is expressed in terms of sine and cosine functions. One form of this identity is \(\sec^{2}(x) = 1 + \tan^{2}(x)\).

• In our exercise, we use this identity to simplify the expression. Transforming \(\sec^{2}(10^{\circ})\) into \(1 + \tan^{2}(10^{\circ})\) makes calculations straightforward.


• The Pythagorean Identity helps in making sense of complex trigonometric equations by enabling us to rewrite them in simpler forms.

This identity essentially helps to connect and transform different trigonometric expressions, making solving questions much easier.

Exact Trigonometric Values

Exact trigonometric values refer to the specific values of trigonometric functions at certain standard angles, like 30°, 45°, 60°, etc. In problems such as this one, finding exact values without a calculator is essential.

• When asked to find exact values without a calculator, recognizing complementary angles and using identities like the Pythagorean Identity becomes invaluable.


• In the context of this exercise, the exact value comes down to recognizing how identities simplify the expression to basic arithmetic steps.

Mastering these values and identities allows you to handle more complex trigonometric expressions and is a useful skill in many mathematical applications.