Reciprocal Identities in Trigonometry (With Examples)

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Reciprocal identities are the reciprocals of the three standard trigonometric functions, namely sine, cosine, and tangent. In trigonometry, reciprocal identities are sometimes called inverse identities. Reciprocal identities are inverse sine, cosine, and tangent functions written as “arc” prefixes such as arcsine, arccosine, and arctan. For instance, functions like sin^-1 (x) and cos^-1 (x) are inverse identities. Either notation is correct and acceptable.

Learning reciprocal identities requires you to be familiar with the formulas of the basic trigonometric identities of sine, cosine, and tangent. These fundamental identities are inverses of the functions. They act as functions reversing the positions of the numerator and denominator. In this case, the sides of the triangle.

The reciprocal for sine function is the cosecant function.

sin (θ) = 1 / csc (θ)

The reciprocal for cosine function is secant function.

cos (θ) = 1 / sec (θ)

The reciprocal for tangent function is cotangent function.

tan (θ) = 1 / cot (θ)

The reciprocal for cosecant function is sine function.

csc (θ) = 1 / sin (θ)

The reciprocal for secant function is cosine function.

sec (θ) = 1 / cos (θ)

The reciprocal for cotangent function is tangent function.

cot (θ) = 1 / tan (θ)

One important thing to note is that all definitions of secant, cosecant, and cotangent involve dividing something that might be zero. For example if cos (x) = 0 then sec(x) is not defined because we’d be dividing by zero. It is why functions like secant, cosecant, and cotangent have holes or gaps in their domains.

Secant

The secant function is the reciprocal of the cosine. It is the hypotenuse ratio to the side adjacent to a given angle in a right triangle. See the figure below to understand the concept more.

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cos (A) = adjacent / hypotenuse = b / c

sec (A) = hypotenuse / adjacent = c / b

Cosecant

The cosecant is the reciprocal of the sine function. The cosecant is the ratio of the hypotenuse of the right triangle to the side opposite a given angle.

sin (A) = opposite / hypotenuse = a / c

csc (A) = hypotenuse / opposite = c / a

Cotangent

The cotangent function is the reciprocal of the tangent. It is the ratio of the adjacent side to the opposite side in a given right triangle.

tan (A) = opposite / adjacent = a / b

cot (A) = adjacent / opposite = b / a

Other Fundamental Trigonometric Identities Applying the Reciprocal Identities

Here are some different important identities or formulas that can be used together with the reciprocal identities when simplifying or verifying identities.

Pythagorean Identities

sin² (θ) + cos² (θ) = 1

tan² (θ) + 1 = sec² (θ)

cot² (θ) + 1 = csc² (θ)

Quotient Identities

tan (θ) = sin (θ) / cos (θ)

cot (θ) = cos (θ) / sin (θ)

Steps for Proving Expressions Involving Reciprocal Identities

When working with trigonometric expressions, it is often desirable to convert one form to an equivalent function that may be more useful. Below are some of the techniques used to establish particular identities. Remember that there is no fixed procedure that works in the proofs for all identities. Nevertheless, you can always proceed with specific steps that will help in many forms.

1. Start manipulating the more complicated side and transform it into more simple trigonometric expressions. Often, you solve the side where you can break down equations into uncomplicated trigonometric functions like sine, cosine, and tangent.
2. Try algebraic operations such as multiplying, factoring, combining and splitting single fractions, and so on.
3. If other steps fail, express each function in terms of sine and cosine functions and then perform appropriate algebraic operations,
4. At each step, keep the other side of the identity in mind. It often reveals what one should do to get there.

See the examples illustrated below to fully understand the reciprocal identities and how they are related to other fundamental identities.

Example 1: Finding Trigonometric Function Values Using the Reciprocal Identities

Find the value of the following expressions using the reciprocal identities.

1. sin (x) = 3 / 7, csc (x) = ?
2. cos (x) = √3 / 2, sec (x) = ?
3. tan (x) = 3, cot (x) = ?
4. sec (x) = π / 5, cos (x) = ?
5. csc (x) = 0.5, sin (x) = ?
6. cot (x) =√2 / 2, tan (x) = ?

Solution

Given the six trigonometric identities, get the reciprocals of each using the reciprocal identities mentioned earlier.

sin (x) = 3 / 7

csc (x) = 7 / 3

cos (x) = √3 / 2

sec (x) = 2 / √3

tan (x) = 3

cot (x) = 1 / 3

sec (x) = π / 5

cos (x) = 5 / π

csc (x) = 0.5 = 1 / 2

sin (x) = 1 / 0.5

sin (x) = 2

cot (x) =√2 / 2

tan (x) = 2 / √2

Final Answer

The reciprocal of sin (x) = 3 / 7 is csc (x) = 7 / 3.

The reciprocal of cos (x) = √3 / 2 is sec (x) = 2 / √3.

The reciprocal of tan (x) = 3 is cot (x) = 1 / 3.

The reciprocal of sec (x) = π / 5 is cos (x) = 5 / π.

The reciprocal of csc (x) = 0.5 is sin (x) = 2.

The reciprocal of cot (x) =√2 / 2 is tan (x) = 2 / √2.

Example 2: Evaluating and Graphing a Secant Equation

Evaluate and graph sec (5π/6).

Solution

Recall the reciprocal identity for cosine and substitute the value 5π/6. Then, graph the function.

sec (5π/6) = 1 / cos (5π/6)

sec (5π/6) = 1 / (-√3 / 2)

sec (5π/6) = - 2 / √3

sec (5π/6) = - 2√3 / 3

Final Answer

The value of sec (5π/6) is - 2√3 / 3.

Example 3: Evaluating and Graphing a Cosecant Function

Evaluate and graph cosecant (7π/6).

Solution

Recall the reciprocal identity for cosine and substitute the value 5π/6.

csc (7π/6) = 1 / sin (7π/6)

csc (7π/6) = 1 / (-1 / 2)

csc (7π/6) = - 2

csc (7π/6) = - 2

Final Answer

The value of csc (7π/6) is - 2.

Example 4: Evaluating a Secant and Tangent Complex Function Using the Reciprocal Identity for Secant

Simplify sec (θ) / tan (θ).

Solution

Shown below is the step-by-step procedure in simplifying the given expression by performing algebraic manipulations. Apply the reciprocal identities by rewriting both the secant and tangent functions in sine and cosine, respectively.

sec(θ) / tan (θ) = [1 / cos (θ)] / [sin (θ) / cos (θ)]

To continue simplifying, invert the functions and multiply.

sec(θ) / tan (θ) = [1 / cos (θ)] [cos (θ) / sin (θ)]

sec(θ) / tan (θ) = 1 / sin (θ)

Recall that one of the reciprocal identities is 1 / sin (θ) equal to csc (θ).

1 / sin (θ) = csc (θ)

It shows that sec(θ) / tan (θ) can be simplified to csc (θ) and established as an identity.

Final Answer

The trigonometric equation sec(θ) / tan (θ) is equal to csc (θ).

Example 5: Proving Identities Using Reciprocal Formulas

Prove the identity sin (x) + cos (x) = [1 + cot (x)] / csc (x).

Solution

Let start simplifying the right side to obtain the more straightforward equation sin (x) + cos (x). The first thing to do is to reduce the fraction. Let us rewrite the cotangent and cosecant functions on the right side of the equation.

[1 + cot (x)] / csc (x) = [1 + (cos (x) / sin (x))] / (1 / sin (x))

Next, divide the fraction. In dividing the fraction, we invert the terms and multiply.

[1 + cot (x)] / csc (x) = [1 + (cos (x) / sin (x))] [sin(x)]

Then, distribute sin (x) to all the terms inside the parentheses and simplify the fraction.

[1 + cot (x)] / csc (x) = sin (x) + [cos (x) sin (x) / sin (x)]

[1 + cot (x)] / csc (x) = sin (x) + cos (x)

Final Answer

Therefore, sin (x) + cos (x) = [1 + cot (x)] / csc (x) is a correct identity.

Example 6: Simplifying Expressions Using the Reciprocal Identities

Simplify the given expression using the reciprocal identities.
cot (x) [sin (x) + tan (x)] / [csc (x) + cot (x)]

Solution

Observe the given equation and identify probable functions to be simplified using the reciprocal identities and quotient identities. Start by replacing those functions with their equivalence. The main goal in the first step is to break down the expressions into the three trigonometric functions, namely sine, cosine, and tangent.

cot (x) [sin (x) + tan (x)] / [csc (x) + cot (x)] = [cos (x) / sin (x) [sin (x) + (sin (x) / cos (x)]] / (1 / sin (x)) + (cos (x) / sin (x))

Next, simplify the equation by using the distribution identity. As you can observe, you can cancel a lot of terms in the numerator.

cot (x) [sin (x) + tan (x)] / [csc (x) + cot (x)] = cos (x) +1 / [ (1 + cos (x) / sin (x))]

cot (x) [sin (x) + tan (x)] / [csc (x) + cot (x)] = cos (x) +1 [sin (x) / 1 + cos (x)]

cot (x) [sin (x) + tan (x)] / [csc (x) + cot (x)] = sin (x)

Final Answer

The simplified form of the complex expression cot (x) [sin (x) + tan (x)] / [csc (x) + cot (x)] is sin (x).

Example 7: Solving for the Values of Cosecant, Secant, and Cotangent Functions

Given the triangle shown below, find the value of csc (A), sec (A), and cot (A).

Solution

Recall that the cosecant is the reciprocal of sine. The cosecant is the ratio of the hypotenuse to the opposite. Referring to the triangle, the value of hypotenuse is equal to 5 while the value of the opposite side is 3.

csc (A) = hypotenuse / opposite

csc (A) = 5 / 3

The secant function is the ratio of the hypotenuse to the adjacent side. The value of the hypotenuse is 5, and the value of the adjacent side is 4. Substitute these values to the equation.

sec (A) = hypotenuse / adjacent

sec (A) =5 / 4

The cotangent is the ratio of the opposite to the adjacent side. The value of the opposite side is three, while the value of the adjacent side is 4.

cot (A) = adjacent / opposite

cot (A) = 4 / 3

Final Answer

The values of csc (A), sec (A), and cot (A) are 5/3, 5/4, and 4/3.

Example 8: Obtaining Values of Cosecant, Secant, and Cotangent Using the Triangle Illustrations

Given the following triangles, identify the values of csc (X), sec (W), and cot (R).

Solution

First, let us solve for csc (X). Remember that cosecant is the ratio of the hypotenuse to the opposite side. From the given triangle, the value of hypotenuse is 10, while the value of the opposite side is 8.

csc (X) = hypotenuse / opposite

csc (X) = 10 / 8

csc (X) = 5 / 4

Next, let us solve the value of sec (W) by looking at the second figure. The secant function is the ratio of the hypotenuse to the adjacent side. The hypotenuse length is equal to 30, while the size of the adjacent side is 18.

sec (W) = hypotenuse / adjacent

sec (W) = 30 / 18

sec (W) = 5 / 3

Finally, let us solve for the value of cot (R). Looking at the third right triangle, the value of the opposite and adjacent sides are 15 and 36. The cotangent is the ratio of the adjacent to the opposite.

cot (R) = adjacent / opposite

cot (R) = 36 / 15

cot (R) = 12 / 5

Final Answer

The values of csc (X), sec (W), and cot (R) are 5/4, 5/3, and 12/5, respectively.

Example 9: Getting the Value of Reciprocals

Given the values of trigonometric functions, solve for the value of their reciprocals.

1. cos (θ) = 0.2, sec (θ) = ?
2. cot (θ) = 4/3, tan (θ) = ?
3. sin (θ) = ⅓, csc (θ) = ?

Solution

These functions are reciprocals. For cos (θ) = 0.2, it is easier to express the value as a fraction. Once narrowed down like a fraction, get the reciprocal to obtain the value of sec (θ).

cos (θ) = 0.2

cos (θ) = 2 / 10

cos (θ) = 1 / 5

sec (θ) = 1 / cos (θ)

sec (θ) = 5 / 1

sec (θ) = 5

cot (θ) = 4 / 3

tan (θ) = 1 / cot (θ)

tan (θ) = 3 / 4

sin (θ) = ⅓

csc (θ) = 1 / sin (θ)

csc (θ) = 3

Final Answer

The value of reciprocal of cos (θ) = 0.2 is 5.

The reciprocal value of cot (θ) = 4 / 3 is 3 / 4.

The value of reciprocal of sin (θ) = ⅓ is 3.

Example 10: Establishing Identities Using the Fundamental Identities

Establish the identity sec (θ) - tan (θ) sin (θ) = cos (θ).

Solution

Generally, we start with the more complicated side and transform it into the other side using fundamental identities, algebra, or different established identities.

sec (θ) - tan (θ) sin (θ) = cos (θ)

sec (θ) - tan (θ) sin (θ) = (1 / cos (θ)) - [sin (θ) / cos (θ)] [sin(θ)]

sec (θ) - tan (θ) sin (θ) = (1 / cos (θ)) - [sin2 (θ) / cos (θ)]

sec (θ) - tan (θ) sin (θ) = [1 - sin2 (θ)] / cos (θ)

sec (θ) - tan (θ) sin (θ) = cos2 (θ) / cos (θ)

sec (θ) - tan (θ) sin (θ) = cos (θ)

Final Answer

The expression sec (θ) - tan (θ) sin (θ) = cos (θ) is a verified identity.

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This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.

© 2021 Ray