Trigonometry is a significant branch of maths that delves into the relationships between the sides and angles of triangles. Fundamental trigonometric identities are pivotal in simplifying and resolving a myriad of trigonometric problems. This section will provide an in-depth exploration of the reciprocal, quotient, and Pythagorean identities.
Reciprocal Identities
The reciprocal identities are derived from the basic trigonometric ratios: sine (sin), cosine (cos), and tangent (tan). These identities introduce three additional trigonometric functions that are reciprocals of the primary functions:
1. Cosecant (csc)
The cosecant is the reciprocal of sine. It is defined as: csc(θ) = 1/sin(θ)
Example Question: Given sin(θ) = 0.4, determine csc(θ).
Solution: Using the definition of cosecant: csc(θ) = 1/0.4 = 2.5
2. Secant (sec)
The secant is the reciprocal of cosine. It is defined as: sec(θ) = 1/cos(θ)
Example Question: If cos(θ) = 0.5, what is sec(θ)?
Solution: Using the definition of secant: sec(θ) = 1/0.5 = 2
3. Cotangent (cot)
The cotangent is the reciprocal of tangent. It is defined as: cot(θ) = 1/tan(θ)
Example Question: Given tan(θ) = 1.5, find cot(θ).
Solution: Using the definition of cotangent: cot(θ) = 1/1.5 = 0.67
Quotient Identities
Quotient identities establish a relationship between the tangent and cotangent functions and the sine and cosine functions. These identities are essential for simplifying trigonometric expressions and solving equations.
1. Tangent
The tangent function is the quotient of sine and cosine. It is defined as: tan(θ) = sin(θ)/cos(θ)
Example Question: If sin(θ) = 0.6 and cos(θ) = 0.8, determine tan(θ).
Solution: Using the quotient identity for tangent: tan(θ) = 0.6/0.8 = 0.75
2. Cotangent
The cotangent function is the reciprocal of tangent. It is also the quotient of cosine and sine. It is defined as: cot(θ) = cos(θ)/sin(θ)
Example Question: Given sin(θ) = 0.5 and cos(θ) = 0.866, compute cot(θ).
Solution: Using the quotient identity for cotangent: cot(θ) = 0.866/0.5 = 1.732
Pythagorean Identities
The Pythagorean identities are derived from the Pythagoras theorem. These identities relate the squares of the sine, cosine, and tangent functions. The three primary Pythagorean identities are:
1. sin2(θ) + cos2(θ) = 1
2. 1 + tan2(θ) = sec2(θ)
3. 1 + cot2(θ) = csc2(θ)
Example Question: If sin(θ) = 0.7, determine cos2(θ).
Solution: Using the first Pythagorean identity: cos2(θ) = 1 - sin2(θ) cos2(θ) = 1 - 0.49 = 0.51
FAQ
The unit circle is a circle with a radius of 1 unit, centred at the origin of a coordinate plane. The trigonometric functions are defined in terms of the unit circle. For any angle θ, the sine is the y-coordinate and the cosine is the x-coordinate of the point where the terminal side of the angle intersects the unit circle. The reciprocal identities are directly related to this concept. For instance, the cosecant, which is the reciprocal of sine, represents the hypotenuse divided by the opposite side (or y-coordinate) on the unit circle. Similarly, the secant and cotangent represent reciprocals of the cosine and tangent functions, respectively, in relation to the unit circle.
Understanding the domains and ranges of trigonometric functions is crucial because it ensures the validity of mathematical operations and prevents undefined values. For example, the cosine function has a range between -1 and 1. If we were to find the secant (which is the reciprocal of cosine) of an angle where the cosine is zero, we would end up with a division by zero, resulting in an undefined value. Similarly, knowing the domain helps in determining the intervals where the function is defined, especially when solving equations or inequalities. Being aware of these domains and ranges ensures that the solutions derived using the identities are valid and meaningful.
Absolutely! The fundamental identities are often the starting point for deriving more complex trigonometric identities. For instance, using the Pythagorean identity, sin2(θ) + cos2(θ) = 1, and the quotient identity for tangent, one can derive the Pythagorean identity for tangent: 1 + tan2(θ) = sec2(θ). Similarly, the double angle and half-angle identities, product-to-sum identities, and sum-to-product identities can all be derived using combinations of the fundamental identities. These derived identities further expand the toolkit available for solving trigonometric problems and proving other mathematical statements.
Trigonometry, underpinned by these fundamental identities, plays a pivotal role in various real-world applications. Engineers, architects, and physicists often use these identities to analyse forces, design structures, and understand wave patterns. For example, electrical engineers use trigonometric identities to analyse alternating current circuits. In navigation, trigonometry helps in determining distances and directions. The fundamental identities simplify complex problems, making them more manageable and solvable. By transforming and simplifying trigonometric expressions using these identities, professionals can derive meaningful insights and solutions for real-world challenges.
The term "fundamental" is used to describe these identities because they form the foundation upon which many other trigonometric concepts, proofs, and applications are built. These identities are derived directly from the basic definitions of trigonometric functions and the Pythagorean theorem. They are essential tools that are frequently used to simplify trigonometric expressions, prove other identities, and solve trigonometric equations. Without a solid understanding of these fundamental identities, it would be challenging to grasp more advanced trigonometric concepts and applications.
Practice Questions
Given sin(α) = 3/5, we can use the Pythagorean identity to find cos(α). Since sin2(α) + cos2(α) = 1, we have cos2(α) = 1 - 9/25 = 16/25. Thus, cos(α) = 4/5.
Using the quotient identity, tan(α) = sin(α)/cos(α) = (3/5) / (4/5) = 3/4.
The reciprocal identities give: csc(α) = 1/sin(α) = 5/3 sec(α) = 1/cos(α) = 5/4 cot(α) = 1/tan(α) = 4/3
In a right triangle, the sum of the acute angles is 90 degrees. Thus, γ = 90 degrees - β.
Using the angle subtraction formula for tangent, we have: tan(90 degrees - β) = sin(90 degrees - β) / cos(90 degrees - β)
Since sin(90 degrees - β) = cos(β) and cos(90 degrees - β) = sin(β), the expression becomes: tan(γ) = cos(β) / sin(β)
Given tan(β) = 2/3, we can deduce that sin(β) = 2/sqrt(13) and cos(β) = 3/sqrt(13).
Substituting these values in, we get: tan(γ) = (3/sqrt(13)) / (2/sqrt(13)) = 3/2.
