Evaluate Sin 135 Without Using A Calculator Brainly

This tool helps you understand how to find the sine of an angle in the second quadrant (90° to 180°) by finding its reference angle. The default shows how to evaluate sin 135 without using a calculator.

What is “Evaluate sin 135 Without Using a Calculator”?

The task to evaluate sin 135 without using a calculator is a common trigonometry problem that tests your understanding of the unit circle, reference angles, and the properties of trigonometric functions in different quadrants. It means finding the exact value of the sine of 135 degrees using geometric principles rather than a digital calculator. This skill is fundamental for students of mathematics and physics, as it builds a deeper intuition for how these functions work. The process relies on recognizing that sin(135°) has the same magnitude as the sine of a “special angle” in the first quadrant.

Anyone studying trigonometry, pre-calculus, or even higher-level physics and engineering will need to know how to evaluate sin 135 without using a calculator. A common misconception is that an exact value cannot be found for angles outside the first quadrant (0°-90°) without a calculator. In reality, by using reference angles, we can relate any angle back to one of the easily-memorized special angles (30°, 45°, 60°).

Formula and Mathematical Explanation

The core principle to evaluate sin 135 without using a calculator is the concept of a reference angle. The reference angle (θ’) is the smallest acute angle that the terminal arm of an angle (θ) makes with the horizontal x-axis. For an angle in Quadrant II (like 135°), the formula is:

θ’ = 180° – θ

Once you find the reference angle, you can find its sine. The sign (+ or -) of the final answer depends on the quadrant. In Quadrant II, the y-values are positive, and since sine corresponds to the y-coordinate on the unit circle, sin(θ) is positive. Therefore, for any angle θ in Quadrant II, sin(θ) = sin(180° – θ) = sin(θ’). This method is crucial when you need to evaluate sin 135 without using a calculator.

Variables Explained

Variables used in evaluating second-quadrant sine.

Variable	Meaning	Unit	Typical Range
θ	The original angle.	Degrees	90° < θ < 180° (for this context)
θ’	The reference angle.	Degrees	0° < θ’ < 90°
sin(θ)	The sine of the original angle.	Ratio (unitless)	0 < sin(θ) < 1 (for this context)

Practical Examples

Example 1: Evaluate sin(150°)

1. Identify Quadrant: 150° is in Quadrant II.
2. Find Reference Angle: θ’ = 180° – 150° = 30°.
3. Evaluate Sine of Reference Angle: We know sin(30°) = 1/2.
4. Determine Sign: Sine is positive in Quadrant II.
5. Conclusion: sin(150°) = 1/2.

Example 2: Evaluate sin(120°)

1. Identify Quadrant: 120° is in Quadrant II.
2. Find Reference Angle: θ’ = 180° – 120° = 60°.
3. Evaluate Sine of Reference Angle: From our knowledge of special triangles, sin(60°) = √3 / 2. This is a key step if you want to evaluate sin 135 without using a calculator or similar angles.
4. Determine Sign: Sine is positive in Quadrant II.
5. Conclusion: sin(120°) = √3 / 2.

For more complex calculations, you might use a Trigonometry Calculator for verification.

How to Use This Calculator

This calculator is designed to visually explain the process required to evaluate sin 135 without using a calculator, and it works for any angle in the second quadrant.

1. Enter an Angle: Type an angle between 90 and 180 degrees into the input field. The calculator will update in real time.
2. Review the Primary Result: The large green box shows the exact value of sin(θ) as a fraction and its decimal approximation.
3. Analyze Intermediate Values: The section below shows you the quadrant, the calculated reference angle, and the formula used to get the result.
4. Explore the Unit Circle Chart: The dynamic SVG chart visualizes the angle you entered. You can see the terminal arm, the main angle (in blue), and the reference angle (in yellow-green), providing a geometric understanding of the calculation. Understanding the Unit Circle is essential for trigonometry.

Key Concepts for Evaluating Trigonometric Functions

Mastering the ability to evaluate sin 135 without using a calculator requires understanding several core concepts:

• Understanding Quadrants: The Cartesian plane is divided into four quadrants. The sign of sine, cosine, and tangent depends on the quadrant. The mnemonic “All Students Take Calculus” helps remember which functions are positive in quadrants I, II, III, and IV, respectively.
• The Unit Circle: A circle with a radius of 1 centered at the origin. It provides a powerful visual tool where the cosine and sine of an angle are the x and y coordinates of the point on the circle, respectively. This is the foundation for being able to evaluate sin 135 without using a calculator.
• Reference Angles: This is the most critical concept. It allows you to relate any angle to an equivalent acute angle in the first quadrant, simplifying the problem immensely. A Reference Angle Calculator can help practice this skill.
• Special Triangles: The 30-60-90 and 45-45-90 triangles provide the exact sine and cosine values for special angles (30°, 45°, 60°). Memorizing these ratios is non-negotiable for solving problems without a calculator.
• The ASTC Rule: Confirms that in Quadrant II (where 135° lies), only the Sine function (and its reciprocal, cosecant) is positive.
• Radians vs. Degrees: While this calculator uses degrees, all these principles apply equally to angles measured in radians. Knowing how to convert between them is a related, vital skill.

Frequently Asked Questions (FAQ)

Why is sin(135°) positive?

Because 135° is in Quadrant II. On the unit circle, all y-coordinates in this quadrant are positive, and sin(θ) corresponds to the y-coordinate.

What is the reference angle for 135°?

The reference angle is 180° – 135° = 45°. This is the key step to evaluate sin 135 without using a calculator.

What is the value of cos(135°)?

Using the same reference angle of 45°, we know cos(45°) = √2 / 2. However, cosine (the x-coordinate) is negative in Quadrant II, so cos(135°) = -√2 / 2. You can check this with a Cosine Calculator.

Can this method be used for other quadrants?

Yes. The formula for the reference angle changes (θ’ = θ – 180° for Q3, θ’ = 360° – θ for Q4), but the principle of using the reference angle and then applying the correct sign based on the quadrant is universal.

Why do we need to learn to do this without a calculator?

It builds a fundamental understanding of trigonometric principles that is essential for calculus and other advanced math and science fields. It teaches you the ‘why’ behind the answer, not just the ‘what’.

What are the “special angles” in trigonometry?

They are 0°, 30°, 45°, 60°, and 90° and their multiples. Their trigonometric values can be expressed as simple fractions and square roots derived from special right triangles. Understanding these is essential to evaluate sin 135 without using a calculator.

What is tan(135°)?

Since tan(θ) = sin(θ)/cos(θ), tan(135°) = (√2 / 2) / (-√2 / 2) = -1. You can explore this further with a Tangent Calculator.

Is sin(135°) the same as sin(-225°)?

Yes. -225° is coterminal with 135° (since -225° + 360° = 135°). They are at the same position on the unit circle and thus have the same sine, cosine, and tangent values.

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