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What is the Value of sin 45 degrees?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
The value of sin 45 degrees is 1/sqrt(2) or approximately 0.707. It represents the ratio of the length of the side opposite to the 45-degree angle to the length of the hypotenuse in a right-angled triangle.
Simple Example
Quick Example
Imagine you are flying a kite and the string makes a 45-degree angle with the ground. If the string is 10 meters long, the height of the kite from the ground (the 'opposite' side) can be found using sin 45 degrees. It helps us calculate heights or distances when we know an angle and one side.
Worked Example
Step-by-Step
Let's find the value of sin 45 degrees using a right-angled isosceles triangle.
---Step 1: Consider a right-angled triangle ABC, where angle B is 90 degrees. Let angles A and C both be 45 degrees (since it's an isosceles right triangle).
---Step 2: In an isosceles right-angled triangle, the two sides forming the right angle are equal. Let AB = BC = 'a' units.
---Step 3: Use the Pythagorean theorem to find the hypotenuse AC. AC^2 = AB^2 + BC^2 = a^2 + a^2 = 2a^2.
---Step 4: So, AC = sqrt(2a^2) = a * sqrt(2) units.
---Step 5: Recall the definition of sine: sin(angle) = Opposite side / Hypotenuse.
---Step 6: For angle C (which is 45 degrees), the opposite side is AB = 'a' and the hypotenuse is AC = a * sqrt(2).
---Step 7: Therefore, sin 45 degrees = AB / AC = a / (a * sqrt(2)).
---Step 8: Cancel out 'a' from the numerator and denominator. sin 45 degrees = 1 / sqrt(2).
---Answer: The value of sin 45 degrees is 1/sqrt(2).
Why It Matters
Understanding sin 45 degrees is crucial in fields like Engineering and Physics to design structures or calculate forces. Space Technology uses it to track satellite orbits, and AI/ML algorithms might use trigonometric functions for image processing. Many engineers, scientists, and even game developers use this concept daily.
Common Mistakes
MISTAKE: Confusing sin 45 degrees with cos 45 degrees or tan 45 degrees. | CORRECTION: Remember that sin 45 = 1/sqrt(2), cos 45 = 1/sqrt(2), and tan 45 = 1. They are different ratios.
MISTAKE: Writing sin 45 degrees as sqrt(2) instead of 1/sqrt(2). | CORRECTION: Always remember it's 'opposite over hypotenuse', so the hypotenuse (the longest side) will be in the denominator, making the value less than 1.
MISTAKE: Forgetting to rationalize the denominator and leaving the answer as 1/sqrt(2) when asked for a simplified form. | CORRECTION: To rationalize, multiply both numerator and denominator by sqrt(2). So, 1/sqrt(2) becomes sqrt(2)/2.
Practice Questions
Try It Yourself
QUESTION: If the hypotenuse of a right-angled triangle is 8 cm and one angle is 45 degrees, what is the length of the side opposite to the 45-degree angle? | ANSWER: 8 * (1/sqrt(2)) = 8/sqrt(2) = 4*sqrt(2) cm
QUESTION: A ladder leans against a wall, making a 45-degree angle with the ground. If the base of the ladder is 5 meters from the wall, what is the length of the ladder? (Hint: Use cos 45 degrees first, then find the hypotenuse) | ANSWER: Length of ladder = 5 / cos 45 = 5 / (1/sqrt(2)) = 5*sqrt(2) meters
QUESTION: In a right-angled triangle, if two angles are 45 degrees and 90 degrees, and the side opposite the 90-degree angle (hypotenuse) is 10 units, find the length of the other two sides. | ANSWER: Each of the other two sides = 10 * sin 45 = 10 * (1/sqrt(2)) = 10/sqrt(2) = 5*sqrt(2) units
MCQ
Quick Quiz
What is the rationalized value of sin 45 degrees?
1/sqrt(2)
sqrt(2)
sqrt(2)/2
2/sqrt(2)
The Correct Answer Is:
C
sin 45 degrees is 1/sqrt(2). To rationalize, multiply numerator and denominator by sqrt(2), which gives sqrt(2)/2. Options A is unrationalized, B and D are incorrect values.
Real World Connection
In the Real World
When ISRO launches satellites, they use complex calculations involving angles and distances. Knowing trigonometric values like sin 45 degrees helps engineers predict trajectories and ensure the satellite reaches its correct orbit. Even drone deliveries, like those planned by companies in India, rely on precise angle calculations for stable flight paths and accurate drop-offs.
Key Vocabulary
Key Terms
SINE: Ratio of opposite side to hypotenuse in a right triangle | HYPOTENUSE: The longest side of a right-angled triangle, opposite the right angle | OPPOSITE SIDE: The side across from a given angle in a right triangle | RIGHT-ANGLED TRIANGLE: A triangle with one angle exactly 90 degrees | RATIONALIZE: To remove square roots from the denominator of a fraction
What's Next
What to Learn Next
Great job understanding sin 45 degrees! Next, you should explore the values of sin for other special angles like 30 degrees and 60 degrees. This will help you build a complete picture of trigonometry and solve more complex problems, preparing you for higher classes.
