Fundamental Trig Identities Page
There are three fundamental trig identities: sine, cosine, and tangent.
There are three reciprocal trig identities: secant, cosecant, and cotangent.
Sine, Cosine & Tangent
You will remember that the formulas for the three basic trig identities are:
\(\text{sine} = \frac{opposite}{hypotenuse}\)
\(\text{cosine} = \frac{adjacent}{hypotenuse}\)
\(\text{tangent} = \frac{opposite}{adjacent}\)
Cosecant, Secant and Cotangent
You will remember that the formulas for the three inverse trig identities are:
\(\text{cosecant} = \frac{hypotenuse}{opposite}\)
\(\text{secant} = \frac{hypotenuse}{adjacent}\)
\(\text{cotangent} = \frac{adjacent}{opposite}\)
Fundamental Trig Identities Quiz
Instructions: Use your knowledge of trig identities to answer the following questions. Then check your answers in the next section.
Triangle XYZ above is a 30-60-90 degree triangle.
Angle X measures 60 degrees.
Side XY is a units in length.
Side YZ is a√3 units in length.Side XZ is 2a units in length.
Question 1: The cosecant of angle X is 2/√3. What is the sine?Question 2: The secant of angle X is 2 . What is the cosine?
Question 3: The cotangent of angle X is 1/√3. What is the tangent?
Question 4: If tan2 X = 3, what is sec2?
Question 5: Prove that sin X × csc X = 1
Trig Identities – Answers
Answer 1:
\(\text{The reciprocal of cosecant is sine} = \frac{1}{csc A} = \text{sin A}\)
So, if the cosecant of angle X is 2/√3 , the sine is √3/2.Answer 2:
\(\text{The reciprocal of secant is cosine} = \frac{1}{sec A} = \text{cos A}\)
So, if the secant of angle X is 2, the cosine is 1/2.
Answer 3:
\(\text{The reciprocal of cotangent is tangent} = \frac{1}{cot A} = \text{tan A}\)
So, if the cotangent of angle X is 1/√3, the tangent is √3.Answer 4:
If tan2 X = 3, what is sec2?
tan2 X + 1 = sec2 X
3 + 1 = sec2
sec2 = 4
Answer 5:
Prove that sin X × csc X = 1
Here, we have a 30-60-90 degree triangle and the relative lengths of the sides are provided in the facts of the problem.
The sine of X is therefore is √3/2 and the cosecant is 2/√3 .Proof: √3/2 × 2/√3 = 2√3/2√3 = 1
Trig Identities – Abbreviations
The following abbreviations are commonly used when discussing trigonometric identities:
- sin = sine
- cos = cosine
- tan = tangent
- csc = cosecant
- sec = secant
- cot = cotangent
The abbreviation may also be used with the identity of the angle to which it relates, such as csc α or cot λ.
Reciprocal Identities – Defined
Secant is the reverse of cosine.
Cosecant is the reverse of sine.
Cotangent is the reverse of tangent.
We can express these identities as fractions that contain 1 in the numerator as shown below:
Cosecant (CSC)
\(\text{The reciprocal of sine is cosecant} = \frac{1}{sin A} = \text{csc A}\)
Secant (Sec)
\(\text{The reciprocal of cosine is secant} = \frac{1}{cos A} = \text{sec A}\)
Cotangent (Cot)
\(\text{The reciprocal of tangent is cotangent} = \frac{1}{tan A} = \text{cot A}\)
Sine (Sin)
\(\text{The reciprocal of cosecant is sine} = \frac{1}{csc A} = \text{sin A}\)
Cosine (Cos)
\(\text{The reciprocal of secant is cosine} = \frac{1}{sec A} = \text{cos A}\)
Tangent (Tan)
\(\text{The reciprocal of cotangent is tangent} = \frac{1}{cot A} = \text{tan A}\)
Reciprocal Identities – Proof
When working out problems on trig identities, you may need to do a proof.
“Doing a proof” means that you need to check your work, using a different formula.
Fundamental Trig Identities – Proofs
In order to prove trig identities, remember the following equations:
sin Α × csc A = 1
cos A × sec A = 1
tan A × cot A = 1
Pythagorean Identities
Pythagorean identities are useful in order to manipulate equations and expressions.
Here are some of the most important trigonometric Pythagorean identities:
sin2 A + cos2 A = 1
tan2 A + 1 = sec2 A
1 + cot2 A = csc2 A
Trig Identities – Differing Notations
You might see different notations when calculating the square of the trig identities.
For ease of reference, the parentheses in trig identities are sometimes removed.
For instance, sin2 A is the same as (sin A)2 and tan2 A is the same as (tan A)2.
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