Find the value of each trigonometric ratio. Make sure you reduce/simplify the fractions. 1) sin z 2) sinc 40 24 36 39 32 3) cos X 4) cos X X 15 Z 37 12 9 12 35 5) tan 2 6) tan C Z 24 40 30 32 50 C X 7) sin A 8) tanz B с 30 40 24 50 For 9) and 10) find ALL SIX trigonometric ratios ( sin, cos, tan, sec, csc, and cot) NOT only the one that is typed. 9) tan 4
The Correct Answer and Explanation is :
To determine the values of the six trigonometric ratios for an angle θ, we need to know the lengths of the sides of a right-angled triangle relative to θ. The six trigonometric functions are:
- Sine (sin θ): Opposite side / Hypotenuse
- Cosine (cos θ): Adjacent side / Hypotenuse
- Tangent (tan θ): Opposite side / Adjacent side
- Cosecant (csc θ): Hypotenuse / Opposite side
- Secant (sec θ): Hypotenuse / Adjacent side
- Cotangent (cot θ): Adjacent side / Opposite side
Given the information provided, it appears that some of the data might be incomplete or unclear. However, based on the available details, let’s attempt to calculate the trigonometric ratios for the following scenarios:
1) sin z
Without specific information about the sides of the triangle or the angle z, it’s not possible to determine sin z. To calculate sin z, we need the length of the side opposite angle z and the hypotenuse.
2) sinc 40 24 36 39 32
This expression is unclear. If it refers to a triangle with sides 24, 36, and 40, we can use the Pythagorean theorem to check if it’s a right-angled triangle:
24² + 36² = 576 + 1296 = 1872 40² = 1600
Since 1872 ≠ 1600, this is not a right-angled triangle, and trigonometric ratios cannot be determined without additional information.
3) cos X
Without details about the sides of the triangle or the angle X, it’s not possible to determine cos X. To calculate cos X, we need the length of the adjacent side and the hypotenuse.
4) cos X X 15 Z 37 12 9 12 35
This expression is unclear. If it refers to a triangle with sides 9, 12, and 15, we can check if it’s a right-angled triangle:
9² + 12² = 81 + 144 = 225 15² = 225
Since 225 = 225, this is a right-angled triangle. For angle X, cos X = Adjacent / Hypotenuse = 12 / 15 = 4 / 5.
5) tan 2
Without information about the sides of the triangle or the angle, it’s not possible to determine tan 2. To calculate tan 2, we need the length of the opposite side and the adjacent side.
6) tan C Z 24 40 30 32 50 C X
This expression is unclear. If it refers to a triangle with sides 24, 32, and 40, we can check if it’s a right-angled triangle:
24² + 32² = 576 + 1024 = 1600 40² = 1600
Since 1600 = 1600, this is a right-angled triangle. For angle C, tan C = Opposite / Adjacent = 24 / 32 = 3 / 4.
7) sin A
Without information about the sides of the triangle or the angle A, it’s not possible to determine sin A. To calculate sin A, we need the length of the opposite side and the hypotenuse.
8) tan B с 30 40 24 50
This expression is unclear. If it refers to a triangle with sides 24, 30, and 50, we can check if it’s a right-angled triangle:
24² + 30² = 576 + 900 = 1476 50² = 2500
Since 1476 ≠ 2500, this is not a right-angled triangle, and trigonometric ratios cannot be determined without additional information.
9) tan 4
Without information about the sides of the triangle or the angle, it’s not possible to determine tan 4. To calculate tan 4, we need the length of the opposite side and the adjacent side.
10) tan 4
This is a repeat of question 9. Without information about the sides of the triangle or the angle, it’s not possible to determine tan 4.
General Approach to Finding Trigonometric Ratios:
- Identify the Triangle Type: Determine if the triangle is a right-angled triangle. If it is, you can use the Pythagorean theorem to find missing sides.
- Label the Sides: Label the sides of the triangle as opposite, adjacent, and hypotenuse relative to the angle of interest.
- Apply Trigonometric Definitions: Use the definitions of the trigonometric functions to find the desired ratios.
- Simplify the Results: Ensure that all fractions are in their simplest form.
For a more detailed explanation and visual demonstration, you might find the following video helpful:
video4 3 Right Triangle Trigonometryturn0search7