Math homework
Unit Circle Homework: Find Exact Values
Solve unit circle homework with reference angles, exact sine and cosine values, quadrant checks and a focused Lirno hint.
Your worksheet asks for the exact sine and cosine of 150°. You remember that 30° belongs somewhere in the calculation, but two decisions remain: which value belongs to cosine, and which one needs a minus sign? Unit circle homework gets much easier when you make those decisions on a sketch before reaching for a table.
Unit circle homework starts with the point
Draw a horizontal axis and a vertical axis through the center of a circle. Give the circle radius one. Start on the right side, where the positive horizontal axis meets the circle. A positive angle turns counterclockwise from that starting point. A negative angle turns clockwise. The endpoint is a point on the circle, and its coordinates in horizontal-then-vertical order are cosine and sine.
For 150°, make a rough turn past the top of the circle and stop before the far left. Your endpoint is left of center and above the horizontal axis. Write “cos negative, sin positive” while the sketch is still in front of you. This step works even if you have forgotten every exact value in the table. You already know that a positive cosine or a negative sine would contradict the point.
Keep the axes and the starting ray on your sketch. A curved arrow by itself can be ambiguous when you look at your work later. If you draw the arrow from the positive horizontal axis, your teacher can see what you treated as zero. For a point on an axis, label that point directly. At 90°, for example, the point is (0, 1), so cosine is zero and sine is one.
Measure the small angle to the horizontal axis
The reference angle is the acute angle between the final ray and the horizontal axis for the non-axis examples in this guide. At 150°, the ray is 30° short of the negative horizontal axis, so the reference angle is 180° − 150° = 30°. If you subtract 90° instead, you get the angle to the vertical axis. That 60° measures a real gap in your sketch, but it is the wrong gap for this method.
You can use the same drawing in radians. A half turn is π and a whole turn is 2π. At 7π/6, the final ray has gone π/6 past the negative horizontal axis, because 7π/6 − 6π/6 = π/6. Now it lies below that axis and left of the center. Both coordinates are negative. Write the subtraction with a common denominator if you are prone to losing the π.
The OpenStax unit circle lesson shows how the special angles give exact coordinates and how the circle extends those values into other quadrants. Compare its diagram with your own drawing. Look for the direction of rotation and the horizontal line from which the reference angle is measured.
Recover the magnitudes from a triangle
For a 30° reference angle, the vertical leg of the first-quadrant reference triangle is the short side. One way to remember its length is to cut an equilateral triangle of side two in half. The half-base has length one, the hypotenuse has length two, and Pythagoras gives the remaining leg as √3. Rescale so that the hypotenuse is one. The short leg becomes 1/2 and the long leg becomes √3/2.
At 30°, the vertical coordinate has magnitude 1/2 and the horizontal coordinate has magnitude √3/2. At 60°, they switch. At 45°, the two legs are equal and each has length √2/2. Draw the acute angle on the triangle if you cannot tell which side is opposite it. The labels “short leg” and “long leg” alone do not tell you which coordinate to use.
Return to the 150° problem. The 30° magnitudes stay in their original horizontal and vertical order. Apply the signs you found on your sketch: cos 150° = −√3/2 and sin 150° = 1/2. For 7π/6, you use the same magnitudes and a different sign pattern: cos(7π/6) = −√3/2 and sin(7π/6) = −1/2. Put each answer on its own line if the worksheet asks for sine first. The ordered pair on your sketch still puts cosine first.
Use Lirno to check the first point of disagreement
If your school permits AI help, capture the complete assignment in Lirno. Include the prompt, the units, any diagram and the instruction to give exact values. Before asking for help, compare what the app read with the paper. For example, 7π/6, −7π/6 and 7π/8 are different inputs. A clear photo can still be misread. Retake it or correct the input before using any hint.
Try placing the ray yourself. If you are unsure about the position, ask for a hint tied to that single decision: “I think 7π/6 is π/6 past a half turn. Which quadrant does that put it in? Leave sine and cosine for me.” Use the hint to make your own sketch, then finish the values from your reference triangle. If you already finished, ask for a check of one line at a time and compare any explanation with your sketch.
A plausible explanation can contain an error. Verify the final coordinates yourself against the ray and the reference angle, and follow your teacher’s rules for what you may submit. Lirno cannot grant permission or guarantee correctness, grades or mastery. If the school restricts uploading assignments, practise with an example you wrote yourself on a separate page or use the paper method.
Check the answer in two different ways
First check direction. At 150°, the point is closer to the left end of the horizontal axis than to the top of the vertical axis. The horizontal magnitude should therefore exceed the vertical magnitude. That rules out the tempting pair (−1/2, √3/2), even though its signs look right.
Then check the circle equation. Squaring your coordinates should give values that add to one. For (−√3/2, 1/2) that gives 3/4 + 1/4 = 1. Both signs disappear when you square, and addition gives the same sum if you swap the coordinates. Use this test to catch a magnitude or copying error. Use the diagram to catch a sign or order error.
If your teacher permits a calculator check, match its angle mode to the prompt. You can compare the decimal to the exact value you wrote without replacing the exact value with a rounded number. When you get an unexpected decimal, check the angle mode and the input before erasing a sound derivation. The exam error log offers a way to record whether the slip was in the units, position, magnitudes or answer order.
Try a new angle without the worked example
Cover the previous answers and find the point for −π/3. Start on the right and rotate clockwise one sixth of a turn. The point is below the horizontal axis and to the right of the center. The reference angle is π/3. Its cosine is 1/2 and its sine is −√3/2. If you ended up above the axis, try it again with an arrow that starts on the positive horizontal axis.
Next try 2π/3. Its reference angle is also π/3, but it ends above the horizontal axis and to the left. That makes the coordinates (−1/2, √3/2). Compare the two sketches. The magnitudes match because both small angles are π/3. The signs differ because the points are in opposite quadrants. In a third problem, use π/2 so you have to notice an axis angle instead of automatically looking for a small triangle.
Once you can do those three problems, mix them with a conversion question and one problem in degrees. The interleaved math practice guide shows how to mix prompts so that you have to choose what to do first. For this homework, write one cue beside each answer: “clockwise,” “axis point,” “past π,” or “degrees.” If you can explain the cue and reconstruct the coordinate order the next day, you have a way to recover the value even when the table slips your mind.
Translate the sketch into a homework explanation
When the prompt asks you to justify an answer, give the reader the same information that made your sketch useful. For 150°, you might write that the ray ends in the second quadrant and is 30° from the negative horizontal axis. Then name the first-quadrant coordinates at 30° and explain that reflection across the vertical axis changes the horizontal sign. This gives a reason for each part of the answer and lets someone check the signs without trying to interpret an imprecise drawing.
For 7π/6, you can describe the point as a half turn beyond π/6. A half turn takes a point to the opposite side of the center, which reverses both coordinates. This explanation and the reference-angle method give the same result. Try writing both once. The comparison helps when a later worksheet specifically asks for symmetry and you already know how to do the numerical calculation.
If your diagram and written explanation disagree, keep both visible until you find the first mismatch. Suppose your diagram places 5π/3 above the horizontal axis. Rewrite it as 2π − π/3 and draw the full turn before stepping back clockwise. The endpoint then sits below the axis. Record the corrected position first and calculate the values only after that. A correction that shows what changed will be easier to use when you review the assignment.