Pls teach me how to evaluate sin 375° without using calculator
375°= 360° + 15°
sin ( 360° + θ ) = sin θ
sin 375° = sin ( 360° + 15° ) = sin 15°
15° = 60° - 45°
sin ( A + B ) = sin ( A ) cos ( B ) - cos ( A ) sin ( B )
sin 15° = sin ( 60° - 45° ) = sin ( 60° ) cos ( 45° ) - cos ( 60° ) sin ( 45° )
sin 15° = ( √3 / 2 ) ∙ ( √2 / 2 ) - ( 1 / 2 ) ∙ ( √2 / 2 )
sin 15° = √3 ∙ √2 / 2 ∙ 2 - 1 ∙ √2 / 2 ∙ 2
sin 15° = √3 ∙ √2 / 2 ∙ 2 - √2 / 2 ∙ 2
sin 15° = ( √3 ∙ √2 - √2 ) / 2 ∙ 2
sin 15° = √2 ∙ ( √3 - 1 ) / 2 ∙ 2
sin 15° = √2 ∙ ( √3 - 1 ) / √2 ∙ √2 ∙ 2
sin 15° = ( √3 - 1 ) / √2 ∙ 2
sin 15° = ( √3 - 1 ) / 2√2
My typo.
sin ( A - B ) = sin ( A ) cos ( B ) - cos ( A ) sin ( B )
To evaluate sin 375° without using a calculator, we can make use of the unit circle and some trigonometric identities. Here's a step-by-step process:
1. Start by converting the given angle to a reference angle within the range of 0° to 360°. To do this, subtract or add multiples of 360° until you get an angle within this range.
In this case, since 375° is greater than 360°, we subtract 360°:
375° - 360° = 15°
2. In the unit circle, draw a horizontal line from the origin (0,0) to the point where the angle intersects the unit circle. This line represents the cosine value of the angle.
3. Draw a vertical line from the point of intersection to the x-axis. This line represents the sine value of the angle.
4. Since 15° is within the first quadrant of the unit circle, both the sine and cosine values are positive.
5. Calculate the value of the sine by looking at the vertical line you drew in step 3. It will intersect the x-axis at a certain point. Find out the y-coordinate of the intersection point.
Since the radius of the unit circle is 1, and the y-coordinate of the point of intersection is also the sine value, the sine of 15° is equal to the y-coordinate of the intersection point.
6. To determine this value, you can use trigonometric ratios of special angles. The angle of 15° is not a special angle, but we can use the fact that sin (90° - θ) = cos θ to our advantage.
In this case, we can rewrite sin 15° as sin (90° - 75°), since 15° + 75° equals 90°.
7. Using the identity sin (90° - θ) = cos θ, we can evaluate sin (90° - 75°) as cos 75°.
8. Repeat steps 2, 3, and 4 for the angle of 75° to find the cosine value.
9. Finally, substitute the value of cos 75° as the value of sin 15° since they are equal by the identity sin (90° - θ) = cos θ.
10. Therefore, sin 375° is equal to sin 15°, which is equal to cos 75°.
Note: In step 8, you may need to use trigonometric identities and special angles to evaluate cos 75°.
I hope this helps! Let me know if you have any further questions.
Sure! To evaluate sin 375° without using a calculator, we can use the unit circle and trigonometric identities.
Step 1: Understand the periodicity of sine function
The sine function repeats itself every 360 degrees. So, if we subtract multiples of 360 from 375°, we can find an equivalent angle in the first revolution.
375° - 360° = 15°
Step 2: Identify a related angle
To find a related angle, we look for an angle on the unit circle that has the same reference angle (in this case, 15°). In the first revolution, both 15° and 345° have the same reference angle of 15°.
Step 3: Use the symmetry property
Since sine is an odd function, sin x = -sin (-x). Therefore,
sin 345° = -sin 15°
Step 4: Evaluate sin 15°
We can use the 15°-75°-90° special right triangle to find sin 15°. In this triangle, the ratios are:
sin 15° = opposite side / hypotenuse
sin 15° = 1 / sqrt(6)
Step 5: Calculate sin 345° using symmetry
Since -sin 15° = -1 / sqrt(6),
sin 345° = -1 / sqrt(6)
So, sin 375° = -1 / sqrt(6)