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103 LearnersLast updated on September 26, 2025
Math Formula for Sin, Cos, and Tan
In trigonometry, the primary functions are sine (sin), cosine (cos), and tangent (tan). These functions relate the angles and sides of a right triangle. In this topic, we will learn the formulas for sin, cos, and tan.
List of Math Formulas for Sin, Cos, and Tan
The primary trigonometric functions are sine, cosine, and tangent. Let’s learn the formula to calculate sin, cos, and tan.
Math formula for Sin
The sine of an angle in a right triangle is the ratio of the length of the opposite side to the length of the hypotenuse.
It is calculated using the formula: sin(θ) = opposite/hypotenuse
Math formula for Cos
The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the length of the hypotenuse.
It is calculated using the formula: cos(θ) = adjacent/hypotenuse
Math formula for Tan
The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side.
It is calculated using the formula: tan(θ) = opposite/adjacent
Importance of Sin, Cos, and Tan Formulas
In math and real life, we use the sin, cos, and tan formulas to analyze and solve problems involving right triangles.
Here are some important points about sin, cos, and tan:
- These functions help in calculating angles and sides in trigonometry.
- By learning these formulas, students can easily understand concepts like wave functions, oscillations, and even some aspects of calculus.
- To find distances or heights indirectly, we often use these trigonometric ratios.
Tips and Tricks to Memorize Sin, Cos, and Tan Math Formulas
Students often find trigonometry formulas tricky and confusing.
Here are some tips and tricks to master the sin, cos, and tan formulas:
- Use simple mnemonics like SOH-CAH-TOA, which stands for Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
- Connect the use of sin, cos, and tan with real-life scenarios like ramps, ladders, or shadows.
- Use flashcards to memorize the formulas and rewrite them for quick recall, and create a formula chart for quick reference.
Common Mistakes and How to Avoid Them While Using Sin, Cos, and Tan Math Formulas
Students make errors when calculating sin, cos, and tan.
Here are some mistakes and the ways to avoid them, to master them.
Mistake 1
Confusing the sides of the triangle
Students sometimes mix up the opposite, adjacent, and hypotenuse sides.
To avoid this error, always identify the angle in question and label the sides accordingly before applying the formulas.
Mistake 2
Using incorrect angle measurements
When using trigonometric functions, ensure that the angle is measured in the correct unit (degrees or radians).
Double-check the problem to use the appropriate unit.
Mistake 3
Forgetting the relationship between functions
Students often forget the identity relationships like tan(θ) = sin(θ)/cos(θ).
Remembering these relationships can simplify complex problems.
Mistake 4
Incorrect calculator settings
Ensure your calculator is set to the correct mode (degree or radian) as per the problem requirements to avoid incorrect answers.
Mistake 5
Assuming results without calculation
Students sometimes assume the results without solving the problem.
Always perform the calculations and verify the answer with the context of the problem.
Examples of Problems Using Sin, Cos, and Tan Math Formulas
Problem 1
What is the sin of an angle θ if the opposite side is 3 and the hypotenuse is 5?
The sin(θ) is 0.6
Explanation
To find sin(θ), use the formula: sin(θ) = opposite/hypotenuse = 3/5 = 0.6
Problem 2
Find the cos of an angle θ if the adjacent side is 4 and the hypotenuse is 5?
The cos(θ) is 0.8
Explanation
To find cos(θ), use the formula: cos(θ) = adjacent/hypotenuse = 4/5 = 0.8
Problem 3
What is the tan of an angle θ if the opposite side is 5 and the adjacent side is 12?
The tan(θ) is 0.4167
Explanation
To find tan(θ), use the formula: tan(θ) = opposite/adjacent = 5/12 ≈ 0.4167
Problem 4
A ladder leans against a wall, making an angle of 60° with the ground. If the ladder is 10 meters long, what is the height it reaches on the wall?
The height is 8.66 meters
Explanation
Using sin(60°) = opposite/hypotenuse = height/10 Height = 10 * sin(60°) = 10 * 0.866 = 8.66 meters
Problem 5
A ramp is inclined at 30° to the horizontal. If the length of the ramp is 5 meters, what is the horizontal distance covered by the ramp?
The horizontal distance is 4.33 meters
Explanation
Using cos(30°) = adjacent/hypotenuse = horizontal distance/5 Horizontal distance = 5 * cos(30°) = 5 * 0.866 = 4.33 meters
FAQs on Sin, Cos, Tan Math Formulas
1.What is the formula for sin?
2.What is the formula for cos?
3.How to find tan?
4.What is the tan of an angle if the opposite side is 7 and the adjacent side is 24?
5.What is the sin of 45°?
Glossary for Sin, Cos, and Tan Math Formulas
- Sine (sin): A trigonometric function that represents the ratio of the opposite side to the hypotenuse of a right triangle.
- Cosine (cos): A trigonometric function that represents the ratio of the adjacent side to the hypotenuse of a right triangle.
- Tangent (tan): A trigonometric function that represents the ratio of the opposite side to the adjacent side of a right triangle.
- Hypotenuse: The longest side of a right triangle, opposite the right angle.
- Trigonometry: A branch of mathematics that studies relationships involving lengths and angles of triangles.
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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.
