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Math  /  Trigonometry

QuestionWolkstuet on Trigonometric functions
1. Fill the Blank space (a) 2π2 \pi radians equals \qquad - degrues (b) 1 radians equars \qquad clesrees
Q π\pi rodians equais \qquad degrees (d) For any circle, one rotation. \qquad desnees. (2. Convert each radian miasure to desvees a) π6\frac{\pi}{6} b) π3\frac{\pi}{3} d) 7π10\frac{7 \pi}{10} c). −3π4-\frac{3 \pi}{4} e). 2π2 \pi (3) Convert each degree to radian Measure a). 30∘30^{\circ} c) 15∘15^{\circ} b). 120∘120^{\circ} d) 360∘360^{\circ}
4. Usins pythagorean identity if sin⁡θ=35\sin \theta=\frac{3}{5} and θ\theta is in the second quadrant, fand cos⁡θ\cos \theta.
5. If cos⁡t=2423\cos t=\frac{24}{23} and tt is in the fourth quadrant, find 1.6. What is the demain and Vanpe of sine and Consin sint. functions. T. Clase the Six trisono Metrec functic which one quadsant is tre in Which queadtant is - Ve.

Studdy Solution

STEP 1

1. 2π2\pi radians equals 360∘360^{\circ}.
2. 11 radian equals 180∘π\frac{180^{\circ}}{\pi}.
3. π\pi radians equals 180∘180^{\circ}.
4. One full rotation in a circle equals 360∘360^{\circ}.
5. To convert radians to degrees, use the formula: degrees=radians×180∘π\text{degrees} = \text{radians} \times \frac{180^{\circ}}{\pi}.
6. To convert degrees to radians, use the formula: radians=degrees×π180∘\text{radians} = \text{degrees} \times \frac{\pi}{180^{\circ}}.
7. The Pythagorean identity is sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1.
8. The sine and cosine functions have a domain of all real numbers and a range of [−1,1][-1, 1].
9. The values of the six trigonometric functions depend on the quadrant in which the angle lies.

STEP 2

1. Fill in the blanks for conversion between radians and degrees.
2. Convert each radian measure to degrees.
3. Convert each degree measure to radians.
4. Use the Pythagorean identity to find cos⁡θ\cos \theta given sin⁡θ\sin \theta and the quadrant.
5. Find the trigonometric values given the cosine and the quadrant.
6. Determine the domain and range of sine and cosine functions.
7. Identify the signs of the six trigonometric functions in different quadrants.

STEP 3

Fill in the blank: 2π radians equals 360∘2\pi \text{ radians equals } 360^{\circ}

STEP 4

Fill in the blank: 1 radian equals 180∘π1 \text{ radian equals } \frac{180^{\circ}}{\pi}

STEP 5

Fill in the blank: π radians equals 180∘\pi \text{ radians equals } 180^{\circ}

STEP 6

Fill in the blank: One full rotation equals 360∘\text{One full rotation equals } 360^{\circ}

STEP 7

Convert π6\frac{\pi}{6} radians to degrees: π6×180∘π=30∘\frac{\pi}{6} \times \frac{180^{\circ}}{\pi} = 30^{\circ}

STEP 8

Convert π3\frac{\pi}{3} radians to degrees: π3×180∘π=60∘\frac{\pi}{3} \times \frac{180^{\circ}}{\pi} = 60^{\circ}

STEP 9

Convert 7π10\frac{7\pi}{10} radians to degrees: 7π10×180∘π=126∘\frac{7\pi}{10} \times \frac{180^{\circ}}{\pi} = 126^{\circ}

STEP 10

Convert −3π4-\frac{3\pi}{4} radians to degrees: −3π4×180∘π=−135∘-\frac{3\pi}{4} \times \frac{180^{\circ}}{\pi} = -135^{\circ}

STEP 11

Convert 2π2\pi radians to degrees: 2π×180∘π=360∘2\pi \times \frac{180^{\circ}}{\pi} = 360^{\circ}

STEP 12

Convert 30∘30^{\circ} to radians: 30∘×π180∘=π630^{\circ} \times \frac{\pi}{180^{\circ}} = \frac{\pi}{6}

STEP 13

Convert 15∘15^{\circ} to radians: 15∘×π180∘=π1215^{\circ} \times \frac{\pi}{180^{\circ}} = \frac{\pi}{12}

STEP 14

Convert 120∘120^{\circ} to radians: 120∘×π180∘=2π3120^{\circ} \times \frac{\pi}{180^{\circ}} = \frac{2\pi}{3}

STEP 15

Convert 360∘360^{\circ} to radians: 360∘×π180∘=2π360^{\circ} \times \frac{\pi}{180^{\circ}} = 2\pi

STEP 16

Use the Pythagorean identity sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1 to find cos⁡θ\cos \theta given sin⁡θ=35\sin \theta = \frac{3}{5} and θ\theta is in the second quadrant.
Since sin⁡θ=35\sin \theta = \frac{3}{5}, we have: (35)2+cos⁡2θ=1 \left(\frac{3}{5}\right)^2 + \cos^2 \theta = 1 925+cos⁡2θ=1 \frac{9}{25} + \cos^2 \theta = 1 cos⁡2θ=1−925 \cos^2 \theta = 1 - \frac{9}{25} cos⁡2θ=1625 \cos^2 \theta = \frac{16}{25} Since θ\theta is in the second quadrant, cos⁡θ\cos \theta is negative: cos⁡θ=−45 \cos \theta = -\frac{4}{5}

STEP 17

Given cos⁡t=2425\cos t = \frac{24}{25} and tt is in the fourth quadrant, find sin⁡t\sin t. Use the Pythagorean identity: sin⁡2t+cos⁡2t=1 \sin^2 t + \cos^2 t = 1 sin⁡2t+(2425)2=1 \sin^2 t + \left(\frac{24}{25}\right)^2 = 1 sin⁡2t+576625=1 \sin^2 t + \frac{576}{625} = 1 sin⁡2t=1−576625 \sin^2 t = 1 - \frac{576}{625} sin⁡2t=49625 \sin^2 t = \frac{49}{625} Since tt is in the fourth quadrant, sin⁡t\sin t is negative: sin⁡t=−725 \sin t = -\frac{7}{25}

STEP 18

Determine the domain and range of the sine function: Domain: (−∞,∞) \text{Domain: } (-\infty, \infty) Range: [−1,1] \text{Range: } [-1, 1]

STEP 19

Determine the domain and range of the cosine function: Domain: (−∞,∞) \text{Domain: } (-\infty, \infty) Range: [−1,1] \text{Range: } [-1, 1]

STEP 20

Identify the signs of the six trigonometric functions in different quadrants: - Quadrant I: All functions are positive. - Quadrant II: Sine and cosecant are positive. - Quadrant III: Tangent and cotangent are positive. - Quadrant IV: Cosine and secant are positive.

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