Problem 68 - Algebra 2

Question

Use an angle in standard position to find the exact value of \(\left[\sin \left(-135^{\circ}\right)\right]^{2}+\left[\cos \left(-135^{\circ}\right)\right]^{2} .\) Show your work.

Step-by-Step Solution

Answer

The result of adding the square of \(\sin(-135^{\circ})\) and the square of \(\cos(-135^{\circ})\) is 1.

Step 1: Find the exact values of sine and cosine

We first need to find \(\sin(-135^{\circ})\) and \(\cos(-135^{\circ})\). Recognize that these angles are located in the 3rd quadrant where both sine and cosine are negative. Using the reference angle of 45 degrees, the exact values are \(-\sqrt{2}/2\) and \(-\sqrt{2}/2\) respectively.

Step 2: Square the exact values

Next, we square \(\sin(-135^{\circ})\) and \(\cos(-135^{\circ})\) individually. This gives us \(\left(-\frac{\sqrt{2}}{2}\right)^2 = \frac{1}{2}\) for both sine and cosine.

Step 3: Add the squared values

Finally, add the squared values: \(\frac{1}{2} + \frac{1}{2}\).

Key Concepts

Sine and Cosine Functions

The sine and cosine are fundamental trigonometric functions. They help determine the coordinates of a point on the unit circle that corresponds to a given angle. Trigonometric functions, like sine and cosine, are essential in the study of triangles and circles, and they form the basis for more complex mathematical concepts.

• Sine Function (sin): The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse.
• Cosine Function (cos): The cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse.

In the context of the unit circle (radius = 1), sine and cosine represent the y and x coordinates, respectively. When addressing angles like \(-135^{\circ}\), both sine and cosine take specific values based on the position of the angle on the circle. This leads us to understanding reference angles and their use in solving for trigonometric values more efficiently.

Reference Angles

Reference angles simplify the calculation of trigonometric functions by reducing an angle to a smaller equivalent angle within the first quadrant, where equations are easier to handle.The reference angle for any angle \(\theta\) is the positive acute angle formed between the terminal side of \(\theta\) and the x-axis. For example, the reference angle of \(-135^{\circ}\) is \45^{\circ}\. This is because \(-135^{\circ}\) lies in the third quadrant, and you subtract \180^{\circ}\ from it to find the smallest equivalent positive angle.Using reference angles, we calculate trig function values by knowing equations:

• If \(\theta_{ref} = 45^{\circ}\), both \sin(45^{\circ}) = \frac{\sqrt{2}}{2}\ and \cos(45^{\circ}) = \frac{\sqrt{2}}{2}\ apply.

Remember, the result also takes into account the sign, which varies depending on the quadrant the original angle starts from.

Quadrants in Trigonometry

Understanding quadrants is crucial for correctly assigning signs to trigonometric function outcomes. The Cartesian coordinate system is divided into four quadrants, which help determine whether sine, cosine, and other trig functions are negative or positive.- **First Quadrant**: All trigonometric functions are positive. - **Second Quadrant**: Sine is positive, while cosine and tangent are negative.- **Third Quadrant**: Both sine and cosine are negative, while tangent is positive.- **Fourth Quadrant**: Cosine is positive, while sine and tangent are negative.When considering the angle \-135^{\circ}\, it's positioned in the third quadrant. Hence, both \(\sin(-135^{\circ})\) and \(\cos(-135^{\circ})\) are negative. This contributes importantly to the solution of exercises involving these functions, especially when adding squared values, which are then converted into positive terms.