Grade 10
  1. Number Systems  1.1. Real Numbers  1.1.1. Properties of Real Numbers  1.1.2. Rational and Irrational Numbers  1.1.3. Decimal Representation of Real Numbers  1.1.4. Operations on Real Numbers  1.1.5. Real Numbers on the Number Line  1.2. Euclid's Division Lemma  1.3. Prime Factorization  1.4. LCM and HCF  1.5. Exponents and Radicals  1.5.1. Laws of Exponents  1.5.2. Simplifying Radical Expressions  1.5.3. Scientific Notation
  2. Algebra  2.1. Polynomials  2.1.1. Definition and Types of Polynomials  2.1.2. Degree of a Polynomial  2.1.3. Zeros of a Polynomial  2.1.4. Factorization of Polynomials  2.1.5. Algebraic Identities  2.2. Linear Equations in Two Variables  2.2.1. Graph of Linear Equations  2.2.2. Solutions of Linear Equations  2.2.3. Application of Linear Equations in Word Problems  2.3. Quadratic Equations  2.3.1. Standard Form of a Quadratic Equation  2.3.2. Methods of Solving Quadratic Equations  2.3.2.1. Factorization Method  2.3.2.2. Completing the Square Method  2.3.2.3. Quadratic Formula  2.3.3. Nature of Roots  2.4. Arithmetic Progressions  2.4.1. Definition of Arithmetic Progression  2.4.2. General Term of an AP  2.4.3. Sum of First n Terms of an AP  2.5. Introduction to Functions  2.5.1. Definition and Types of Functions  2.5.2. Domain and Range  2.5.3. Composition of Functions  2.5.4. Inverse of a Function
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Distance Formula  3.3. Section Formula  3.4. Area of a Triangle  3.5. Slope of a Line  3.6. Equation of a Line  3.6.1. Slope-Intercept Form  3.6.2. Point-Slope Form  3.6.3. Two-Point Form  3.6.4. Intercept Form  3.6.5. General Form of a Line
  4. Trigonometry  4.1. Trigonometric Ratios  4.1.1. Sine Cosine Tangent and their Reciprocals  4.1.2. Values of Trigonometric Ratios at Specific Angles  4.2. Trigonometric Identities  4.3. Trigonometric Ratios of Complementary Angles  4.4. Heights and Distances  4.4.1. Angle of Elevation and Depression  4.4.2. Applications in Real-Life Problems
  5. Geometry  5.1. Triangles  5.1.1. Congruence of Triangles  5.1.2. Criteria for Congruence  5.1.3. Properties of Triangles  5.2. Similarity  5.2.1. Similar Figures  5.2.2. Criteria for Similarity  5.2.3. Similarity of Triangles  5.2.4. Applications of Similarity in Real-Life  5.3. Circles  5.3.1. Properties of a Circle  5.3.2. Tangent to a Circle  5.3.3. Number of Tangents from a Point  5.3.4. Angle Subtended by an Arc  5.4. Constructions  5.4.1. Construction of Similar Triangles  5.4.2. Tangent to a Circle  5.4.3. Construction of Angle Bisectors
  6. Mensuration  6.1. Areas of Plane Figures  6.1.1. Area of a Triangle  6.1.2. Area of a Parallelogram  6.1.3. Area of a Trapezium  6.1.4. Area of a Rhombus  6.2. Surface Areas and Volumes  6.2.1. Surface Area of a Cube Cuboid and Cylinder  6.2.2. Surface Area of a Cone and Sphere  6.2.3. Volume of a Cube Cuboid and Cylinder  6.2.4. Volume of a Cone and Sphere  6.3. Conversion of Units  6.4. Frustum of a Cone
  7. Statistics  7.1. Introduction to Statistics  7.2. Collection of Data  7.3. Presentation of Data  7.3.1. Tabular Form  7.3.2. Graphical Form  7.3.2.1. Bar Graph  7.3.2.2. Histogram  7.3.2.3. Frequency Polygon  7.3.2.4. Ogive  7.4. Measures of Central Tendency  7.4.1. Mean Median and Mode  7.4.2. Quartiles and Percentiles  7.5. Measures of Dispersion  7.5.1. Range and Interquartile Range  7.5.2. Standard Deviation and Variance
  8. Probability  8.1. Introduction to Probability  8.2. Experimental Probability  8.3. Theoretical Probability  8.4. Probability of Simple Events  8.5. Complementary Events  8.6. Probability of Compound Events

Grade 10 → Trigonometry → Trigonometric Ratios ↓


Values of Trigonometric Ratios at Specific Angles


Trigonometry is a fascinating branch of mathematics that deals with the relationships between the sides and angles of triangles. One of the fundamental aspects of trigonometry is understanding the values of trigonometric ratios at specific angles. These specific angles often include 0°, 30°, 45°, 60°, and 90°, etc. By understanding these values, we can solve a variety of problems involving triangles in mathematics.

Basics of trigonometric ratios

Before we discuss the values of trigonometric ratios at specific angles, let's revisit the basic trigonometric ratios. These ratios are defined with reference to a right-angled triangle with one angle of 90 degrees. There are three main trigonometric ratios:

  • Sine (( sin )) : This ratio compares the length of the side opposite the angle to the hypotenuse.
  • Cosine (( cos )) : This ratio compares the length of the adjacent side of an angle to the hypotenuse.
  • Tangent (( tan )) : This ratio compares the length of the opposite side to the adjacent side.

The relations can be summarized by the following formulas:

    (sin(theta) = frac{text{opposite}}{text{hypotenuse}}) (cos(theta) = frac{text{adjacent}}{text{hypotenuse}}) (tan(theta) = frac{text{opposite}}{text{adjacent}})

Visualizing these ratios with a unit circle

The unit circle is a powerful tool for understanding trigonometric ratios. The unit circle is simply a circle with a radius of 1. It helps to visualize angles and their corresponding sine, cosine, and tangent values.

θ

In the unit circle given above, the radius is 1. Let the red line make an angle ( theta ) with the positive x-axis. The basic trigonometric ratios in terms of unit circle coordinates will be:

    (sin(theta)) = y-coordinate of the point on the circle (cos(theta)) = x-coordinate of the point on the circle (tan(theta) = frac{sin(theta)}{cos(theta)})

Specific angles and their trigonometric ratios

Now, let us look at the values of trigonometric ratios at some typical angles commonly found in problems.

Angle 0°

At an angle of 0°, the point on the unit circle is (1, 0).

  • (sin(0°) = 0)
  • (cos(0°) = 1)
  • (tan(0°) = 0)

In the context of a right triangle, at 0°, the opposite side is zero (the angle does not open up width), resulting in a sine of 0.

Angle 30°

At a 30° angle, trigonometric values are obtained by dividing an equilateral triangle in half, creating a 30-60-90 triangle.

  • (sin(30°) = frac{1}{2})
  • (cos(30°) = frac{sqrt{3}}{2})
  • (tan(30°) = frac{1}{sqrt{3}})
30°

The right triangle shows ratios, with the vertical line being half the hypotenuse and representing the sine of 30°.

Angle 45°

The values at 45° come from an isosceles right triangle, where the two non-hypotenuse sides are equal.

  • (sin(45°) = frac{sqrt{2}}{2})
  • (cos(45°) = frac{sqrt{2}}{2})
  • (tan(45°) = 1)
45°

In the above diagram, the portions of the unit circle of both sine and cosine are divided equally, resulting in the same value.

Angle 60°

At 60°, the right triangle with 30-60-90 angles still applies, but the sides switch places relative to the angle on the circle.

  • (sin(60°) = frac{sqrt{3}}{2})
  • (cos(60°) = frac{1}{2})
  • (tan(60°) = sqrt{3})
60°

From the above view, (sin(60°)) is longer this time, which shows a different projection on the y-axis.

Angle 90°

Finally, at a 90° angle, the values become particularly distinct.

  • (sin(90°) = 1)
  • (cos(90°) = 0)
  • (tan(90°)) is undefined
90°

Here, the sine value represents the point touching the y-axis maximum, and the cosine is zero as the x-axis does not exist there.

Examples and exercises

Let's look at some examples to apply this knowledge of trigonometric ratios to specific angles.

Example 1: Calculating height

Suppose a ladder rests against a wall and is at an angle of 30° to the ground. If the ladder is 10 m long, how high up the wall does the ladder reach?

We use the sine ratio:

    (sin(30°) = frac{text{opposite}}{text{hypotenuse}} = frac{height}{10}) ) Solving gives: (frac{1}{2} = frac{height}{10} Rightarrow height = 5) meters

Example 2: Finding the length

Imagine a flagpole that casts a shadow of 15 m directly above the sun, creating an angle of elevation of 45°. Calculate the height of the flagpole.

The tangent ratio will be helpful here:

    (tan(45°) = frac{text{opposite}}{text{adjacent}} = frac{height}{15}) Solving gives: (1 = frac{height}{15} Rightarrow height = 15) meters

Exercises for practice

  • Calculate the cosine of a 60° angle in a right triangle with hypotenuse 20 m.
  • Find the sine of the angle where the adjacent side is 8 m, and the hypotenuse is 16 m.
  • If the angle of elevation of the top of a tower from a point is 30° and the distance is 50 m, then find the height of the tower.

Conclusion

Understanding the values of trigonometric ratios at specific angles is important for solving trigonometric problems in mathematics. The relationship of sides to each other in a triangle through sine, cosine, and tangent enables us to find unknown values and create more complex geometric formulas. The visuals and examples provided aim to simplify these concepts and enhance understanding. Practice using these principles in different contexts for a clear understanding of trigonometric ratios and their vast applications.


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Sine Cosine Tangent and their Reciprocals
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Trigonometric Identities

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Values of Trigonometric Ratios at Specific Angles

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