Question

Use the definition or identities to find the exact value of each of the remaining five trigonometric functions of the acute angle \(\theta\). $$\sin \theta=\frac{\sqrt{2}}{2}$$

Short answer

\(\cos \theta = \frac{\sqrt{2}}{2}\), \(\tan \theta = 1\), \(\text{csc} \theta = \sqrt{2}\), \(\text{sec} \theta = \sqrt{2}\), \(\text{cot} \theta = 1\)

Step-by-step solution

Identify \(\theta\)'s reference angle

Recognize that the given value \(\frac{\sqrt{2}}{2}\) is a common trigonometric ratio. For \(\theta\) to be in an acute angle, check if it matches the known trigonometric values for common angles like 45 degrees or \(\pi/4\) radians. \(\theta = 45\) degrees or \(\theta = \pi/4\) radians.

Use trigonometric identities for corresponding angles

Since \(\theta = 45\) degrees is established, use the trigonometric values for 45 degrees to find the remaining functions. \(\theta\) in radians is \(\frac{\pi}{4}\).

Calculate cosine

Using the trigonometric identity \(\text{cosine}\), solve for \(\text{cos} \theta = \frac{1}{\text{hypotenuse}}\). \(\cos \theta = \frac{\sqrt{2}}{2}\)

Calculate tangent

Using the trigonometric identity \(\text{tangent}\), solve for \(\text{tan} \theta = \frac{\sin \theta}{\cos \theta}\). Thus, \(\tan \theta = 1\).

Calculate cosecant (reciprocal of sine)

Using the definition of cosecant, solve for \(\text{csc} \theta = \frac{1}{\sin \theta}\). Hence, \(\text{csc} \theta = \sqrt{2}\).

Calculate secant (reciprocal of cosine)

Using the definition of secant, solve for \(\text{sec} \theta = \frac{1}{\cos \theta}\). Therefore, \(\text{sec} \theta = \sqrt{2}\).

Calculate cotangent (reciprocal of tangent)

Using the definition of cotangent, solve for \(\text{cot} \theta = \frac{1}{\tan \theta}\). Hence, \(\text{cot} \theta = 1\).

Key concepts

sine

The sine function, denoted as \(\text{sin} \theta\), is a fundamental trigonometric function. In a right-angled triangle, the sine of an angle \(\theta\) is defined as the ratio of the length of the opposite side to the hypotenuse. Mathematically, this is expressed as: $$\text{sin} \theta = \frac{\text{opposite}}{\text{hypotenuse}}$$.
For example, in the given exercise, we have \(\text{sin} \theta = \frac{\text{√2}}{2}\) for \(\theta = 45^\text{°}\) or \(\theta = \frac{\text{π}}{4}\) radians.
This is one of the common values that helps in identifying the sine function easily for special angles.

cosine

The cosine function, denoted as \(\text{cos} \theta\), is another basic trigonometric function. It is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle: $$\text{cos} \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$$.
In the context of the given problem, we calculate \(\text{cos} \theta\) for \(\theta = 45^\text{°}\) using known trigonometric values, resulting in $$\text{cos} \theta = \frac{\text{√2}}{2}$$. This value is derived from the unit circle and is particularly useful in solving various trigonometric equations.

tangent

The tangent function, represented as \(\text{tan} \theta\), is the ratio of the sine function to the cosine function. In a right-angled triangle, it is defined as: $$\text{tan} \theta = \frac{\text{opposite}}{\text{adjacent}}$$.
However, using trigonometric identities, it can also be written as: $$\text{tan} \theta = \frac{\text{sin} \theta}{\text{cos} \theta}$$.
For \(\theta = 45^\text{°}\), we have ##\text{tan} \theta = \frac{\frac{\text{√2}}{2}}{\frac{\text{√2}}{2}} = 1$$.
This value is significant because it is one of the simplest ratios in trigonometry.

cosecant

Cosecant, or \(\text{csc} \theta\), is the reciprocal of the sine function. It is defined as: $$\text{csc} \theta = \frac{1}{\text{sin} \theta}$$.
In our example, where \(\text{sin} \theta\) is \(\frac{\text{√2}}{2}\), we get: $$\text{csc} \theta = \frac{1}{\frac{\text{√2}}{2}} = \text{√2}$$.
Understanding cosecant helps in comprehending the relationships between trig functions and their reciprocals.

secant

The secant function, noted as \(\text{sec} \theta\), is the reciprocal of the cosine function: $$\text{sec} \theta = \frac{1}{\text{cos} \theta}$$.
Using our given value for cosine when \(\theta = 45^\text{°}\), which is \(\frac{\text{√2}}{2}\), we calculate: $$\text{sec} \theta = \frac{1}{\frac{\text{√2}}{2}} = \text{√2}$$.
Secant values are especially useful in more advanced trigonometric transformations and real-world engineering applications.

cotangent

Cotangent, represented as \(\text{cot} \theta\), is the reciprocal of the tangent function: $$\text{cot} \theta = \frac{1}{\text{tan} \theta}$$.
It can also be defined as the ratio of the cosine function to the sine function: $$\text{cot} \theta = \frac{\text{cos} \theta}{\text{sin} \theta}$$.
For \(\theta = 45^\text{°}\), with \(\text{tan} \theta = 1\), we find: $$\text{cot} \theta = \frac{1}{1} = 1$$.
Cotangent plays a role in various inverse trigonometric problems, making it essential to understand its reciprocal relationship.

trigonometric identities

Trigonometric identities are fundamental equations involving trigonometric functions that hold true for all angle measures.
Some key identities include:

• Pythagorean Identity: $$\text{sin}^2 \theta + \text{cos}^2 \theta = 1$$
• Reciprocal Identities: $$\text{csc} \theta = \frac{1}{\text{sin} \theta}$$, $$\text{sec} \theta = \frac{1}{\text{cos} \theta}$$, $$\text{cot} \theta = \frac{1}{\text{tan} \theta}$$

These identities help simplify trigonometric expressions and solve complex equations.

acute angle

An acute angle is an angle that measures less than 90 degrees.
In trigonometry, acute angles are commonly used to define trigonometric functions for right-angled triangles.
In the exercise, \(\theta\) as 45 degrees is an acute angle which is useful in identifying and computing trigonometric function values.
Understanding acute angles is crucial for determining the various sine, cosine, and tangent values in right-angled triangles.

45 degrees

45 degrees is a special angle in trigonometry because its trigonometric function values are often used in problems.
For this particular angle:

• $$\text{sin} 45^\text{°} = \frac{\text{√2}}{2}$$
• $$\text{cos} 45^\text{°} = \frac{\text{√2}}{2}$$
• $$\text{tan} 45^\text{°} = 1$$

These values, derived from the unit circle, provide a base for solving trigonometric equations and understanding more about angle properties.

radians

Radians are a unit of measure for angles used extensively in trigonometry.
One complete revolution around a circle is equal to \(\text{2π}\) radians.
For instance, 45 degrees can be converted to radians as: $$\theta = 45^\text{°} \times \frac{\text{π}}{180^\text{°}} = \frac{\text{π}}{4}$$.
Understanding radians helps in transitioning from degrees to the trigonometric circle, facilitating advanced mathematical computations.