Question

Use Fundamental Identities and/or the Complementary Angle Theorem to find the exact value of each expression. Do not use a calculator. $$\frac{\cos 13^{\circ}}{\sin 77^{\circ}}$$

Short answer

1

Step-by-step solution

Identify complementary angles

Recognize that complementary angles add up to 90 degrees. Here, note that 13° and 77° are complementary because 13° + 77° = 90°.

Apply the Complementary Angle Theorem

According to the Complementary Angle Theorem, \(\frac{\text{cos}\theta}{\text{sin}(90^\text{\text{°}} - \theta)} = 1.\) In this problem, \( \theta = 13^\text{\text{°}} \), and \(\text{sin}(90^\text{\text{°}} - 13^\text{\text{°}}) = \text{sin} 77^\text{°}\).

Simplify using the theorem

Substitute the values into the theorem equation: \(\frac{\text{cos} 13^\text{\text{°}}}{\text{sin} 77^\text{°}} = 1\).

Key concepts

trigonometric identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the included variables. They are vital tools in simplifying expressions and solving trigonometric equations.

Some fundamental trigonometric identities include:

• Pythagorean identities
• Reciprocal identities
• Quotient identities
• Co-function (complementary angle) identities

Using these identities can make complex trigonometric problems more manageable. For instance, the Complementary Angle Theorem is one such identity that states that \(\frac{\text{cos}\theta}{\text{sin}(90^\text{\text{°}} - \theta)} = 1.\). This identity helps simplify the given problem by recognizing that angles summing up to 90 degrees have specific relationships between their sine and cosine values. For the exercise given, using the fact that 13° and 77° are complementary allows us to greatly simplify the expression.

complementary angles

Complementary angles are two angles whose measures add up to 90 degrees. This geometric concept is very useful in trigonometry. For example, if \( \theta \) is an angle, then its complement is \(90^\text{\text{°}} -\theta \).

An important trigonometric property of complementary angles is that the cosine of one angle is equal to the sine of its complement. In other words, \( \text{cos} \theta = \text{sin}(90^\text{\text{°}} - \theta) \).

In the exercise, the angles 13° and 77° add up to 90°, making them complementary. By applying the Complementary Angle Theorem, we use the complementary angle relationship to simplify the trigonometric expression effectively.

• \textbf{Theta, θ}: 13°
• \textbf{Complement}: 77° (since 90° - 13° = 77°)

This relationship allows us to transform the given trigonometric expression into a simpler form by applying known identities.

simplifying trigonometric expressions

Simplifying trigonometric expressions involves breaking down complex trigonometric functions into simpler, more manageable parts using various identities and theorems. This method often unveils hidden relationships and makes solving trigonometric equations practical.

In the given exercise, the goal is to simplify \( \frac{\text{cos} 13^\text{\text{°}}}{\text{sin} 77^\text{\text{°}}} \). Using the Complementary Angle Theorem, we identified that \(13^\text{\text{°}}\text{ and }77^\text{\text{°}}\) are complementary angles. Hence, \( \text{cos} 13^\text{\text{°}} = \text{sin} 77^\text{\text{°}} \).

By applying these identities, the fraction becomes \( \frac{\text{cos} 13^\text{\text{°}}}{\text{sin} 77^\text{\text{°}}} = \frac{\text{cos} 13^\text{\text{°}}}{\text{cos} 13^\text{\text{°}}}=1\).

Steps for simplifying using complementary identities:

• Identify complementary angles.
• Apply complementary angle identities.
• Simplify the expression using these identities.

This approach helps in transforming complex trigonometric expressions into their simplest form, making it easier to handle and understand.