Problem 33 - Algebra and Trigonometry

Question

Find a cofunction with the same value as the given expression. $$ \csc 25^{\circ} $$

Step-by-Step Solution

Answer

The cofunction with the same value as \( \csc 25^{\circ} \) is \( \sec 65^{\circ} \).

Step 1: Identifying the Relevant Cofunction Identity

In this case, the specified function is the cosecant, and its cofunction is secant, represented by \( \csc(\theta) = \sec(90^{\circ} - \theta) \).

Step 2: Substituting the Given Value into the Identity

For this step, replace \( \theta \) in the identity with 25 degrees: This then results in \( \csc(25^{\circ}) = \sec(90^{\circ} - 25^{\circ}) \).

Step 3: Simplifying the Expression

By performing subtraction on the right side of the equation, we get \( \csc(25^{\circ}) = \sec(65^{\circ}) \).

Key Concepts

Cosecant (csc)

The cosecant (csc) is one of the six primary trigonometric functions. It's defined as the reciprocal of the sine. In other words, for any angle \( \theta \) in a right-angled triangle, the cosecant is the ratio of the length of the hypotenuse to the length of the opposite side:\
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\( \csc(\theta) = \frac{1}{\sin(\theta)} \).\
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Since the sine function never reaches a value higher than 1, the cosecant function deals with values equal to or greater than 1, or less than or equal to -1, as it is the inverse. Learning to recognize when to apply the cosecant function is pivotal, especially when dealing with right-angled triangles and circles within trigonometric contexts.

Secant (sec)

Similarly to cosecant, secant (sec) is another one of the trigonometric functions and acts as the reciprocal of the cosine function. So, for an angle \( \theta \) the secant is expressed as:\
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\( \sec(\theta) = \frac{1}{\cos(\theta)} \).\
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This function is particularly useful when dealing with circles and oscillations. It has an indefinable range for values between -1 and 1 since cosine values within this interval would lead to division by zero. Consequently, secant values fall outside this range, creating a graph with a distinctive 'U' shape. Remembering the secant as the 'flipped over' cosine can be a helpful mnemonic for understanding its properties.

Complementary Angles

Complementary angles are a pair of angles whose sum amounts to 90 degrees. An important property of a right-angled triangle is that the two non-right angles within it are always complementary.\
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This characteristic is central to the cofunction identities in trigonometry, which imply that the trigonometric functions for complementary angles are related. For example, the sine of an angle is equal to the cosine of its complement:\
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• \( \sin(\theta) = \cos(90^\circ - \theta) \)
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• \( \cos(\theta) = \sin(90^\circ - \theta) \)
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Understanding this relationship allows students to easily transition between trigonometric functions when given a particular angle—or its complement.

Trigonometric Identities

Trigonometric identities are equations that hold true for any angle \( \theta \), allowing for simplification and transformation of trigonometric expressions. The cofunction identities are a fundamental set of these identities. They establish the relationship between function values of complementary angles:\
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• \( \sin(\theta) = \cos(90^\circ - \theta) \)
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• \( \cos(\theta) = \sin(90^\circ - \theta) \)
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• \( \tan(\theta) = \cot(90^\circ - \theta) \)
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• \( \cot(\theta) = \tan(90^\circ - \theta) \)
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• \( \csc(\theta) = \sec(90^\circ - \theta) \)
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• \( \sec(\theta) = \csc(90^\circ - \theta) \)
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Employing these identities makes it substantially easier to work with complex trigonometric functions and solve problems, as demonstrated in the cofunction identity used to equate \( \csc(25^\circ) \) with \( \sec(65^\circ) \) which are complementary angles.