Question

Find the values of the trigonometric functions of an angle of \(-510^{\circ}\).

Short answer

The values of the trigonometric functions for an angle of -510 degrees are: \(\sin(-510^{\circ}) = -\frac{1}{2}\) \(\cos(-510^{\circ}) = -\frac{\sqrt{3}}{2}\) \(\tan(-510^{\circ}) = \frac{1}{\sqrt{3}}\) \(\cot(-510^{\circ}) = \sqrt{3}\) \(\sec(-510^{\circ}) = -\frac{2\sqrt{3}}{3}\) \(\csc(-510^{\circ}) = -2\)

Step-by-step solution

Find the reference angle in the first rotation

To find the reference angle, we need to add or subtract multiples of 360 degrees until the angle is in the range of 0 to 360 degrees: \(-510^{\circ}+360^{\circ} \times 2 = -510^{\circ}+720^{\circ}=210^{\circ}\) So, the reference angle in the first rotation is 210 degrees.

Find the trigonometric functions values for the reference angle

Since 210 degrees is in the 3rd quadrant, we know that sine and cosine functions are negative and tangent function is positive in this quadrant. We can find the trigonometric functions values using the reference angle in the second quadrant (180 - 210 = -30 degrees): \(\sin(-30^{\circ}) = -\frac{1}{2}\) \(\cos(-30^{\circ}) = -\frac{\sqrt{3}}{2}\) \(\tan(-30^{\circ}) = \frac{\sin(-30^{\circ})}{\cos(-30^{\circ})} = \frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}}\)

Find the remaining trigonometric functions values

Using the trigonometric functions values found in Step 2, we can find the values of the other trigonometric functions: \(\cot(-30^{\circ}) = \frac{1}{\tan(-30^{\circ})} = \frac{1}{\frac{1}{\sqrt{3}}} = \sqrt{3}\) \(\sec(-30^{\circ}) = \frac{1}{\cos(-30^{\circ})} = \frac{2}{-\sqrt{3}} = -\frac{2\sqrt{3}}{3}\) \(\csc(-30^{\circ}) = \frac{1}{\sin(-30^{\circ})} = \frac{1}{-\frac{1}{2}} = -2\)

Final Answer

The values of the trigonometric functions for an angle of -510 degrees are: \[\sin(-510^{\circ}) = -\frac{1}{2}\] \[\cos(-510^{\circ}) = -\frac{\sqrt{3}}{2}\] \[\tan(-510^{\circ}) = \frac{1}{\sqrt{3}}\] \[\cot(-510^{\circ}) = \sqrt{3}\] \[\sec(-510^{\circ}) = -\frac{2\sqrt{3}}{3}\] \[\csc(-510^{\circ}) = -2\]

Key concepts

Reference Angle

Understanding the reference angle is crucial in trigonometry when working with angles that are not within the first full circle (0 to 360 degrees). A reference angle is the positive acute angle that forms when the terminal side of the given angle meets the x-axis. This helps us simplify complex angles by using trigonometric identities established in the first quadrant.

To find the reference angle for \(-510^{\circ}\), we add multiples of \(360^{\circ}\) until it lies between \(0^{\circ}\) and \(360^{\circ}\). Here, we find:

• \(-510^{\circ} + 720^{\circ} = 210^{\circ}\)

Thus, the reference angle for \(-510^{\circ}\) is essentially the angle \(210^{\circ}\) in the standard position on the coordinate plane.

Quadrants

The coordinate plane is divided into four sections, called quadrants, which help us determine the sign of trigonometric functions. Quadrants I through IV are arranged counter-clockwise:

• Quadrant I: All trigonometric functions are positive.
• Quadrant II: Sine is positive, cosine and tangent are negative.
• Quadrant III: Tangent is positive, sine and cosine are negative.
• Quadrant IV: Cosine is positive, sine and tangent are negative.

Understanding which quadrant an angle is in is important because it affects the sign (+ or -) of the function's output. For \(210^{\circ}\), we see it lies in Quadrant III, thus, sine and cosine are negative, while tangent is positive.

Sine

The sine (\(\sin\)) function represents the y-coordinate of the point on the unit circle corresponding to a given angle from the x-axis. In the context of our exercise:

• The reference angle is \(210^{\circ}\), which is the same as \(-30^{\circ} \) from the second quadrant's perspective.
• For \(\sin(-30^{\circ})\), it's known that it’s value is \(-\frac{1}{2}\).

Remember, since \(210^{\circ}\) lies in Quadrant III, the sine value retains the negative sign due to that quadrant's properties.

Cosine

The cosine (\(\cos\)) function gives the x-coordinate of an angle's corresponding point on the unit circle. It helps us understand how far left or right the point is:

• For our problem, \(\cos(-30^{\circ})\) results in \(-\frac{\sqrt{3}}{2}\).

Like sine, cosine's sign is determined by the quadrant. As \(210^{\circ}\) is in Quadrant III, cosine is negative in this region. Knowing these quadrant rules helps accurately evaluate trigonometric functions using reference angles.

Tangent

The tangent (\(\tan\)) function is the ratio of the sine to the cosine of an angle. This is helpful since it combines insights from both functions.

• For \(\tan(-30^{\circ})\), we have \(\frac{\sin(-30^{\circ})}{\cos(-30^{\circ})} = \frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}}\).

Unlike sine and cosine, tangent is positive in Quadrant III, aligning with our earlier observation for \(210^{\circ}\). Reviewing these properties allows for a deeper understanding of how angles in each quadrant interact with trigonometric functions.