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Quizzes > Mathematics > Geometry

Master Trigonometric Ratios of Special Angles - Take the Quiz!

Put your skills to the test in our special angles trigonometry quiz!

Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper cut art of trig ratios including sine, cosine, tangent, protractor on golden yellow background.

Ready to level up with our quiz trigonometric ratios? This special angles trigonometry quiz helps students and math enthusiasts master sine, cosine & tangent at 30°, 45° and 60°. Dive into targeted trigonometric ratios practice to reinforce special angle sine cosine tangent patterns and boost confidence in trigonometry special angles quiz questions. With instant feedback and clear explanations, you'll pinpoint areas for improvement and conquer trigonometry problems quiz situations effortlessly. Try a quick trig ratios practice exercise and then tackle a fun quiz on unit circle . Test your skills, track your progress, and ace the challenge - start now!

What is sin 30°?
1/2
?3/2
1
?2/2
The sine of 30° corresponds to the y-coordinate on the unit circle at that angle, which equals 1/2. In a 30-60-90 right triangle, the side opposite the 30° angle is half the hypotenuse. This is a fundamental special-angle value in trigonometry. For more details, see Unit Circle Table.
What is cos 60°?
1/2
-1/2
0
?3/2
Cosine of 60° is the x-coordinate on the unit circle at 60°, which is 1/2. In the 30-60-90 triangle, the adjacent side to the 60° angle is half the hypotenuse. Recognizing these standard values is essential for solving many trig problems. See Unit Circle Table.
What is tan 45°?
-1
?3
0
1
Tangent is the ratio of sine over cosine. At 45°, both sine and cosine equal ?2/2, so their ratio is 1. This makes tan 45° one of the simplest special-angle values. For a refresher, visit Unit Circle Table.
What is sin 90°?
-1
1
?2/2
0
On the unit circle, the point at 90° has coordinates (0, 1), so sin 90° equals 1. This value is fundamental and often used as a boundary in trigonometry. Remembering that sine represents the y-coordinate helps with all angle evaluations. More info at Unit Circle Table.
What is cos 0°?
0
?3/2
1
-1
At 0° on the unit circle, the coordinates are (1, 0), so cos 0° equals 1. This is a key starting point for understanding cosine as the x-coordinate. It's one of the simplest values in trigonometry. See Unit Circle Table.
What is tan 0°?
1
0
?3/3
Undefined
Tangent is sine over cosine. At 0°, sin 0° is 0 and cos 0° is 1, so tan 0° = 0. This makes tangent zero at the origin angle. You can explore further at Unit Circle Table.
What is sin 45°?
?3/3
1/2
?2/2
?3/2
On the unit circle, the point at 45° has coordinates (?2/2, ?2/2), so sin 45° = ?2/2. This value arises from an isosceles right triangle where the legs are equal. Memorizing it helps with many trig computations. More at Unit Circle Table.
What is cos 45°?
1/2
?2/2
-?2/2
?3/2
Cosine of 45° corresponds to the x-coordinate on the unit circle at that angle, which is ?2/2. In a 45-45-90 triangle, each leg is ?2/2 of the hypotenuse. This standard value is crucial for solving many trigonometric problems. See Unit Circle Table.
What is tan 30°?
0
?3/3
?3
1/2
Tangent of 30° is the ratio sin 30°/cos 30°. Since sin 30° = 1/2 and cos 30° = ?3/2, tan 30° = (1/2)/(?3/2) = 1/?3 = ?3/3. Recognizing this ratio is helpful for many geometry problems. Learn more at Unit Circle Table.
What is sin 60°?
?2/2
?3/3
1/2
?3/2
At 60°, the sine (y-coordinate) of the unit circle is ?3/2. This corresponds to the longer leg of a 30-60-90 triangle. Memorizing these special angles is critical for quick calculations. See Unit Circle Table.
What is cos 120°?
1/2
-1/2
-?3/2
?3/2
120° lies in Quadrant II where cosine is negative. The reference angle is 60°, so cos 120° = -cos 60° = -1/2. Understanding signs in each quadrant is essential. More at Unit Circle Table.
What is sin 210°?
-1/2
?3/2
-?3/2
1/2
210° is in Quadrant III where sine is negative. The reference angle is 30°, so sin 210° = -sin 30° = -1/2. Knowing quadrant rules helps determine the sign. More at Unit Circle Table.
What is tan 135°?
-?3
1
?3
-1
135° is in Quadrant II where sine is positive and cosine is negative, making tangent negative. The reference angle is 45°, so tan 135° = -tan 45° = -1. Quadrant sign rules are key here. Visit Unit Circle Table.
What is cos 150°?
?3/2
1/2
-?3/2
-1/2
150° is in Quadrant II where cosine is negative. Its reference angle is 30°, so cos 150° = -cos 30° = -?3/2. Familiarity with reference angles cuts down on calculation time. See Unit Circle Table.
What is sin 330°?
-?3/2
-1/2
1/2
?3/2
330° is in Quadrant IV where sine is negative. The reference angle is 30°, so sin 330° = -sin 30° = -1/2. Recognizing this pattern is essential for quick evaluations. More at Unit Circle Table.
What is tan 300°?
-1/?3
?3
-?3
0
300° is in Quadrant IV where tangent is negative. The reference angle is 60°, so tan 300° = -tan 60° = -?3. Using reference angles and quadrant rules ensures accuracy. See Unit Circle Table.
What is the exact value of sin45°cos30° - cos45°sin30°?
(?2 + ?6)/2
(?6 - ?2)/4
(?6 + ?2)/4
(?2 - ?6)/4
This expression uses the sine difference identity: sin(A - B) = sinA cosB - cosA sinB. Plugging in A = 45° and B = 30° gives sin15° = (?6 - ?2)/4. Knowing compound-angle formulas helps evaluate nonstandard angles. Learn more at Trig Identities.
What is the exact value of cos15°?
(?6 + ?2)/2
(?2 - ?6)/4
(?6 + ?2)/4
(?6 - ?2)/4
Cosine half-angle formula for 30° gives cos15° = ?((1 + cos30°)/2) = ?((1 + ?3/2)/2) = (?6 + ?2)/4. Half-angle identities are powerful for deriving exact values. Read more at Half-Angle Formulas.
Evaluate tan75° exactly.
?3 + 1
2 + ?3
?3 - 1
2 - ?3
Using the tangent addition formula tan(A + B) with A=45° and B=30°: tan75° = (tan45°+tan30°)/(1?tan45°tan30°) = (1 + 1/?3)/(1 ? 1·1/?3) = 2 + ?3. Compound-angle formulas yield exact values. More at Tangent Identities.
What is sin(5?/6)?
-1/2
-?3/2
1/2
?3/2
5?/6 is 150°, which lies in Quadrant II where sine is positive. The reference angle is ?/6 (30°), so sin(5?/6) = sin30° = 1/2. Recognizing radian measures on the unit circle is key. For details, see Unit Circle Table.
What is tan(2?/3)?
-?3
-1/?3
?3
?3/3
2?/3 equals 120°, in Quadrant II where sine is positive and cosine negative, so tangent is negative. The reference angle is ?/3, so tan(2?/3) = -tan(?/3) = -?3. For more, visit Unit Circle Table.
What is the exact value of cos(7?/6)?
-1/2
?3/2
1/2
-?3/2
7?/6 is 210°, located in Quadrant III where cosine is negative. The reference angle is ?/6 (30°), so cos(7?/6) = -cos30° = -?3/2. Understanding radian-quadrant relationships is essential. See Unit Circle Table.
If cos? = -?3/2 and ? is in Quadrant III, what is sin??
1/2
-?3/2
?3/2
-1/2
In Quadrant III both sine and cosine are negative. Given cos? = -?3/2 and using sin²? + cos²? = 1, sin? = -?(1 - (3/4)) = -1/2. Recognizing sign by quadrant is critical. More at Unit Circle Table.
Simplify sec30° - tan30°.
?3
0
2/?3
1/?3
sec30° = 1/cos30° = 2/?3 and tan30° = 1/?3. Their difference is (2/?3) - (1/?3) = 1/?3. Simplifying radicals is a common skill in trig. For more, see Secant Definition.
What is the exact value of sin 18°?
(?5 + 1)/2
(?5 - 1)/4
(?5 + 1)/4
(?5 - 1)/2
Sin 18° is derived from solving a pentagon-based equation or using half-angle identities on 36°, yielding (?5 - 1)/4. This value is less common and requires deeper algebraic manipulation. It appears in problems involving the golden ratio. Read more at Trigonometric Constants.
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Study Outcomes

  1. Identify Special Angle Ratios -

    Recall the exact sine, cosine, and tangent values for 30°, 45°, and 60° with confidence during trigonometric ratios practice.

  2. Apply Ratios to Triangle Problems -

    Use special angle sine, cosine, and tangent values to calculate missing side lengths in right triangles accurately.

  3. Compare Ratio Patterns -

    Analyze how trigonometric ratios change across 30°, 45°, and 60° to recognize underlying fraction and square root patterns.

  4. Interpret Quiz Results -

    Evaluate your performance on the special angles trigonometry quiz to identify strengths and areas for further study.

  5. Memorize Key Values -

    Commit crucial fraction and √-based values to memory for quick recall during math tests or problem-solving sessions.

  6. Boost Trigonometry Confidence -

    Strengthen your understanding of special angles and feel prepared to tackle more advanced trigonometric ratios questions.

Cheat Sheet

  1. Unit Circle Coordinates for 30°, 45°, and 60° -

    Memorize the exact (x,y) pairs on the unit circle - (√3/2, 1/2) at 30°, (√2/2, √2/2) at 45°, and (1/2, √3/2) at 60° - so you can instantly read off cosine (x) and sine (y). These values form the backbone of any special angles trigonometry quiz and are endorsed by academic sources like MIT OpenCourseWare. Practicing these coordinates in a quiz trigonometric ratios setting builds speed and confidence.

  2. 30° - 60° - 90° Triangle Ratios -

    Use the 30° - 60° - 90° right triangle where sides are in the ratio 1:√3:2 (short leg:long leg:hypotenuse) to derive sin 30°=1/2, cos 30°=√3/2, and tan 30°=1/√3. This concrete construction, found in university geometry texts, ensures you grasp why special angle sine, cosine, and tangent take those exact values. Sketching and labeling this triangle before a special angle sine cosine tangent drill is a proven study habit.

  3. 45° - 45° - 90° Triangle Ratios -

    Recall that in a 45° - 45° - 90° triangle, the legs are equal and the hypotenuse is √2 times one leg, giving sin 45°=cos 45°=√2/2 and tan 45°=1. This symmetry is central to many trigonometric ratios practice scenarios and helps you ace the special angles trigonometry quiz questions. Referencing official math curriculum standards confirms its reliability.

  4. SOH-CAH-TOA Mnemonic and Practice -

    Apply SOH-CAH-TOA ("Sine=Opposite/Hypotenuse, Cosine=Adjacent/Hypotenuse, Tangent=Opposite/Adjacent") as your go-to mnemonic for quiz trigonometric ratios on special angles. Pair this with quick mental checks - like "S over H for sugar over honey" - to lock in formulas under timed conditions. Research from educational journals shows mnemonics boost retention by up to 30% in trigonometry learning.

  5. Complementary Angle Relationships -

    Leverage the identity sin θ=cos (90° - θ) and tan θ=cot (90° - θ) to convert between trigonometric ratios and reduce errors on a special angles trigonometry quiz. For example, sin 60° equals cos 30°, which can simplify multi-step questions in a quiz trigonometric ratios challenge. Official exam prep guides recommend using these relationships to cross-verify answers quickly.

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