Sine Cosine And Tangent Formulas Values And How To Solve Problems
Sine, cosine, and tangent (abbreviated as sin, cos, and tan) are three primary trigonometric functions, which relate an angle of a right-angled triangle to the ratios of two sides length. The reciprocals of sine, cosine, and tangent are the secant, the cosecant, and the cotangent respectively. Each of the six trigonometric functions has corresponding inverse functions (also known as inverse trigonometric functions). The trigonometric functions also known as the circular functions, angle functions, or goniometric functions are widely used in all fields of science that are related to Geometry such as navigation, celestial mechanics, solid mechanics, etc.
Read below to know what is a sine function, cosine function, and tangent function in detail.
Sine Cosine Tangent Definition
A right-angled triangle includes one angle of 90 degrees and two acute angles. Each acute angle of a right-angled triangle retains the property of the sine cosine tangent. The sine, cosine, and tangent of an acute angle of a right-angled triangle are defined as the ratio of two of three sides of the right-angled triangle.
As we know, sine, cosine, and tangent are based on the right-angled triangle, it would be beneficial to give names to each of the triangles to avoid confusion.
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“Hypotenuse side” is the longest side.
“Adjacent side” is the side next to angle θ.
“Opposite side” is the side opposite to angle θ.
Accordingly,
Sin θ = Opposite side/Hypotenuse
Cos θ = Adjacent/Hypotenuse
Tan θ = Opposite/Adjacent
What is the Sine Function?
In the right triangle, the sine function is defined as the ratio of the length of the opposite side to that of the hypotenuse side.
Sin θ = Opposite Side/ Hypotenuse Side.
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For example, the sine function of a triangle ABC with an angle θ is expressed as:
Sin θ = a/c
What is the Cosine Function?
In the right triangle, the cosine function is defined as the ratio of the length of the adjacent side to that of the hypotenuse side.
Cos θ = Adjacent Side/Hypotenuse Side
Example:
Considering the figure given above, the cosine function of a triangle ABC with an angle θ is expressed as:
Cos θ = b/c
What is the Tangent Function?
In the right triangle, the tangent function is defined as the ratio of the length of the opposite side to that of the adjacent side.
Tan θ = Opposite Side/Adjacent Side
Example:
Considering the figure given above, the cosine function of a triangle ABC with an angle θ is expressed as:
Tan θ = a/b
Sine Cosine Tangent Table
The values of trigonometric ratios like sine, cosine, and tangent for some standard angles such as 0°, 30°, 45°, 60°, and 90° can be easily determined with the help of the sine cosine tangent table given below. These values are very important to solve trigonometric problems. Hence, it is important to learn the values of trigonometric ratios of these standard angles.
The sine, cosine, and tangent table given below includes the values of standard angles like 0°, 30°, 45°, 60°, and 90°.
Sine, Cosine, and Tangent Table
Did You Know?
Sine and Cosine were introduced by Aryabhatta, whereas the tangent function was introduced by Muhammad Ibn Musa al- Khwarizmi ( 782 CE - 850 CE).
Sine Cosine and Tangent formulas can be easily learned using SOHCAHTOA. As sine is opposite side over hypotenuse side, cosine is adjacent side over hypotenuse side, and tangent is opposite side over the adjacent side.
Solved Examples:
1. Find Cos θ with respect to the following triangle.
Ans: To find Cos θ, we need both adjacent and hypotenuse side.
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The adjacent side in the above triangle is, BC = 8 Cm
But, the hypotenuse side i.e. AC is not given.
To find the hypotenuse side, we use the Pythagoras theorem
AC² = AB² + BC² = 6² + 8² = 100
Hypotenuse side, AC = √100 or 10 cm
Cos θ = Adjacent/Hypotenuse = 8/10
= 4/5
Therefore, Cos θ = 4/5
2. Find the value of Sin 45°, Cos 60°, and Tan 60°.
Solution: Using the trigonometric table above, we have:
Sin 45° = 1/√2
Cos 60° = 1/2
Tan 45°= 1
FAQs on Understanding Sine Cosine And Tangent In Trigonometry
1. What are sine, cosine, and tangent in trigonometry?
Sine, cosine, and tangent are trigonometric ratios that relate the angles of a right triangle to its side lengths. In a right triangle: sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, and tan θ = opposite / adjacent. These ratios help calculate unknown sides or angles and are fundamental in trigonometry, geometry, and physics.
2. What is the formula for sine, cosine, and tangent?
The formulas for sine, cosine, and tangent in a right triangle are sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, and tan θ = opposite/adjacent. These formulas are often remembered using SOH-CAH-TOA. They apply specifically to right-angled triangles and are used to solve trigonometric problems involving angles and sides.
3. How do you use SOH-CAH-TOA?
SOH-CAH-TOA is a mnemonic that helps remember the trigonometric ratios: SOH → sin = opposite/hypotenuse, CAH → cos = adjacent/hypotenuse, and TOA → tan = opposite/adjacent. To use it, identify the given angle and the known sides, choose the correct ratio, substitute the values, and solve for the unknown side or angle.
4. How do you find sine, cosine, and tangent of an angle?
You find sine, cosine, and tangent by dividing the appropriate sides of a right triangle using their definitions. For example, if opposite = 3 and hypotenuse = 5, then sin θ = 3/5 = 0.6. Similarly, divide adjacent by hypotenuse for cosine and opposite by adjacent for tangent. A scientific calculator can also be used for non-right triangle angles.
5. What is the value of sin, cos, and tan for 30°, 45°, and 60°?
The standard trigonometric values are: sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3; sin 45° = √2/2, cos 45° = √2/2, tan 45° = 1; sin 60° = √3/2, cos 60° = 1/2, and tan 60° = √3. These values are commonly used in trigonometry problems and exams.
6. What is the relationship between sine, cosine, and tangent?
The key relationship is tan θ = sin θ / cos θ. Another important identity is the Pythagorean identity: sin²θ + cos²θ = 1. These trigonometric identities connect the three ratios and are widely used to simplify expressions and solve equations.
7. Why is tangent equal to sine divided by cosine?
Tangent equals sine divided by cosine because of their definitions in a right triangle. Since sin θ = opposite/hypotenuse and cos θ = adjacent/hypotenuse, dividing them gives (opposite/hypotenuse) ÷ (adjacent/hypotenuse) = opposite/adjacent = tan θ. This proves the identity tan θ = sin θ / cos θ.
8. How do you find an angle using sine, cosine, or tangent?
You find an angle by using the inverse trigonometric functions: θ = sin⁻¹(value), θ = cos⁻¹(value), or θ = tan⁻¹(value). For example, if sin θ = 0.5, then θ = sin⁻¹(0.5) = 30°. Use a calculator set to degree or radian mode as required.
9. What is the difference between sine and cosine?
The difference between sine and cosine is that sine relates the opposite side to the hypotenuse, while cosine relates the adjacent side to the hypotenuse. In formula form, sin θ = opposite/hypotenuse and cos θ = adjacent/hypotenuse. Both are fundamental trigonometric functions but use different triangle sides.
10. What are sine, cosine, and tangent used for in real life?
Sine, cosine, and tangent are used to calculate angles, heights, distances, and wave patterns in real-life applications. They are applied in engineering, physics, architecture, navigation, and computer graphics. For example, trigonometric ratios help determine the height of a building using angle of elevation and distance measurements.
