108 LearnersLast updated on August 11th, 2025
Trigonometry Formulas
In mathematics, trigonometry is the study of the relationships between the angles and sides of triangles. Trigonometric formulas are essential for solving problems involving angles and distances. In this topic, we will learn the key trigonometry formulas.
List of Trigonometry Formulas
Trigonometry involves various formulas related to the angles and sides of triangles. Let’s learn the essential trigonometry formulas.
Basic Trigonometric Ratios
The basic trigonometric ratios are sine, cosine, and tangent. They are defined as follows for a right-angled triangle:
Sine (sin) of angle θ = Opposite side/Hypotenuse
Cosine (cos) of angle θ = Adjacent side/Hypotenuse
Tangent (tan) of angle θ = Opposite side/Adjacent side
Reciprocal Trigonometric Ratios
The reciprocal trigonometric ratios are cosecant, secant, and cotangent:
Cosecant (csc) of angle θ = 1/sin(θ) = Hypotenuse/Opposite side
Secant (sec) of angle θ = 1/cos(θ) = Hypotenuse/Adjacent side
Cotangent (cot) of angle θ = 1/tan(θ) = Adjacent side/Opposite side
Pythagorean Identities
The Pythagorean identities relate the squares of the basic trigonometric ratios: - sin²(θ) + cos²(θ) = 1 - 1 + tan²(θ) = sec²(θ) - 1 + cot²(θ) = csc²(θ)
Importance of Trigonometry Formulas
In math and real life, trigonometry formulas help solve problems related to angles and distances. Here are some key points:
Trigonometry is used in various fields such as physics, engineering, and architecture.
By learning these formulas, students can easily understand concepts like wave functions, rotations, and oscillations.
Trigonometry helps in calculating heights and distances that are otherwise difficult to measure directly.
Tips and Tricks to Memorize Trigonometry Formulas
Students often find trigonometry formulas challenging. Here are some tips to master them:
Use mnemonics like SOH-CAH-TOA to remember sine, cosine, and tangent ratios.
Relate trigonometry to real-world examples like shadows, ramps, and ladders.
Use flashcards to memorize the formulas and rewrite them for quick recall; create a formula chart for reference.
Common Mistakes and How to Avoid Them While Using Trigonometry Formulas
Students often make errors when using trigonometry formulas. Here are some common mistakes and how to avoid them:
Mistake 1
Confusing Trigonometric Ratios
Students sometimes confuse sine, cosine, and tangent. To avoid this, remember the mnemonic SOH-CAH-TOA:
Sine = Opposite/Hypotenuse
Cosine = Adjacent/Hypotenuse
Tangent = Opposite/Adjacent
Mistake 2
Errors in Applying Reciprocal Ratios
Errors occur when using reciprocal ratios. Always remember:
Cosecant is 1/sine
Secant is 1/cosine
Cotangent is 1/tangent
Mistake 3
Incorrectly Using Pythagorean Identities
Errors in applying Pythagorean identities can occur if not remembered correctly. Double-check: - sin²(θ) + cos²(θ) = 1 - 1 + tan²(θ) = sec²(θ) - 1 + cot²(θ) = csc²(θ)
Mistake 4
Forgetting Angle of Elevation and Depression
Students often forget the concepts of angles of elevation and depression, which are common in trigonometry problems. Remember, the angle of elevation is from the horizontal upward, and the angle of depression is from the horizontal downward.
Mistake 5
Not Using a Calculator Correctly
Inaccurate results often occur due to incorrect calculator use. Ensure that the calculator is in the correct mode (degrees/radians) corresponding to the problem requirements.
Examples of Problems Using Trigonometry Formulas
Problem 1
Find the sine of angle θ in a triangle where the opposite side is 3 and the hypotenuse is 5.
The sine of angle θ is 0.6.
Explanation
Using the sine formula: sin(θ) = Opposite/Hypotenuse = 3/5 = 0.6
Problem 2
Find the cosine of angle θ in a triangle where the adjacent side is 4 and the hypotenuse is 5.
The cosine of angle θ is 0.8.
Explanation
Using the cosine formula: cos(θ) = Adjacent/Hypotenuse = 4/5 = 0.8
Problem 3
A triangle has an opposite side of 7 and an adjacent side of 24. Find the tangent of the angle.
The tangent of the angle is 0.2917.
Explanation
Using the tangent formula: tan(θ) = Opposite/Adjacent = 7/24 ≈ 0.2917
Problem 4
Calculate the secant of angle θ where the adjacent side is 9 and the hypotenuse is 15.
The secant of angle θ is 1.6667.
Explanation
Using the secant formula: sec(θ) = Hypotenuse/Adjacent = 15/9 ≈ 1.6667
Problem 5
In a right triangle, if the opposite side is 12 and the hypotenuse is 13, find the cosecant of the angle.
The cosecant of the angle is 1.0833.
Explanation
Using the cosecant formula: csc(θ) = Hypotenuse/Opposite = 13/12 ≈ 1.0833
FAQs on Trigonometry Formulas
1.What is the sine formula?
2.What is the formula for cosine?
3.How to find the tangent?
4.What is the secant of an angle?
5.What is the cosecant of an angle?
Glossary for Trigonometry Formulas
- Sine: A trigonometric ratio representing the opposite side over the hypotenuse in a right triangle.
- Cosine: A trigonometric ratio representing the adjacent side over the hypotenuse in a right triangle.
- Tangent: A trigonometric ratio representing the opposite side over the adjacent side in a right triangle.
- Secant: The reciprocal of the cosine, representing the hypotenuse over the adjacent side.
- Cosecant: The reciprocal of the sine, representing the hypotenuse over the opposite side.
Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
