SOHCAHTOA Quiz: Defining SOHCAHTOA

CAH in SOHCAHTOA stands for Cosine = Adjacent divided by Hypotenuse. The side adjacent to the angle, not opposite to it, forms the numerator of the cosine ratio. For example, if the adjacent side is 4 and the hypotenuse is 5, then cos(theta) = 4 divided by 5 = 0.8. Option A describes the numerator used in sine. Option C would give a ratio of 1 for any angle, which is not meaningful. Option D is not a side length.

CAH is the middle part of the mnemonic SOHCAHTOA and represents the cosine ratio. Cosine of an angle equals the length of the side adjacent to that angle divided by the length of the hypotenuse. For example, if the adjacent side is 4 and the hypotenuse is 5, then cos(theta) = 4/5 = 0.8. Option A describes the sine ratio. Option C describes the tangent ratio. Option D inverts the cosine formula, giving the secant ratio instead.

tan(theta) = 1 means the opposite side equals the adjacent side. In a right triangle, the two non-right angles are equal when opposite and adjacent are the same length, which occurs at exactly 45 degrees. Option A gives tan(30 degrees) = 1/sqrt(3) approximately 0.577. Option C gives tan(60 degrees) = sqrt(3) approximately 1.732. Option D gives 90 degrees where tangent is undefined because the adjacent side approaches zero.

SOH in SOHCAHTOA stands for Sine = Opposite divided by Hypotenuse. The numerator of the sine ratio is always the side directly across from the angle in question, which is called the opposite side. Option A describes the numerator used in cosine, not sine. Options B and D name trigonometric functions rather than side lengths. Identifying the correct sides for each trigonometric ratio is the foundation of SOHCAHTOA.

The answer is True. Tangent equals opposite divided by adjacent. When the opposite side is longer than the adjacent side, the ratio exceeds 1. This happens for all acute angles greater than 45 degrees. For example, tan(60 degrees) = sqrt(3) which is approximately 1.732, and tan(89 degrees) is an extremely large positive number. Unlike sine and cosine, tangent has no upper bound and increases without limit as the angle approaches 90 degrees.

theta = arcsin(0.6) = 36.87 degrees. This corresponds to the 3-4-5 right triangle where the opposite side is 3 and the hypotenuse is 5, giving sin(theta) = 3/5 = 0.6. Option A gives sin(30 degrees) = 0.5, not 0.6. Option C gives sin(45 degrees) = sqrt(2)/2 approximately 0.707. Option D gives sin(60 degrees) = sqrt(3)/2 approximately 0.866. Only 36.87 degrees satisfies the equation.

TOA in SOHCAHTOA stands for Tangent = Opposite divided by Adjacent. The denominator of the tangent ratio is the adjacent side, which is the side next to the angle that is not the hypotenuse. Option A would describe a different ratio involving the hypotenuse. Option B would give a ratio of 1 since opposite divided by itself is always 1. Option D names a trigonometric function, not a side of the triangle.

The answer is True. Two angles are complementary when they add to 90 degrees. In a right triangle, the two non-right angles always add to 90 degrees. The side opposite one angle is the side adjacent to the other. This means sin(theta) = cos(90 degrees minus theta) for any acute angle. For example, sin(30 degrees) = 0.5 and cos(60 degrees) = 0.5, confirming the relationship holds.

Option A inverts the tangent ratio, giving secant, which is not a standard SOHCAHTOA definition. CAH gives Cosine = Adjacent divided by Hypotenuse, confirming B. TOA gives Tangent = Opposite divided by Adjacent, confirming C. SOH gives Sine = Opposite divided by Hypotenuse, confirming D.

Using the Pythagorean identity, sin squared plus cos squared = 1. Substituting gives sin squared = 1 minus (12/13) squared = 1 minus 144/169 = 25/169. Taking the positive square root gives sin(theta) = 5/13. This corresponds to a 5-12-13 right triangle where the opposite side is 5 and the hypotenuse is 13. Option B inverts the fraction. Options C and D do not satisfy the identity.

The answer is True. TOA in SOHCAHTOA stands for Tangent = Opposite divided by Adjacent. In a right triangle, this ratio compares the side directly across from the angle to the side next to it, excluding the hypotenuse. For example, if the opposite side is 3 and the adjacent side is 4, then tan(theta) = 3 divided by 4 = 0.75. This is distinct from sine and cosine, which both involve the hypotenuse.

SOH is the first part of the mnemonic SOHCAHTOA and represents the sine ratio. Sine of an angle equals the length of the side opposite that angle divided by the length of the hypotenuse. For example, in a right triangle where the opposite side is 3 and the hypotenuse is 5, sin(theta) = 3 divided by 5 = 0.6. Options B and D describe cosine, and option C describes tangent.

tan(theta) = 3/4 means the opposite side is 3 and the adjacent side is 4. Using the Pythagorean theorem, the hypotenuse equals the square root of (9 plus 16) = the square root of 25 = 5. Therefore sin(theta) = opposite divided by hypotenuse = 3 divided by 5 = 3/5. This is the 3-4-5 right triangle. Option B gives the cosine. Options C and D invert or mix up the side lengths.

sin(theta) = 3/5 means the opposite side is 3 and the hypotenuse is 5. Using the Pythagorean theorem, the adjacent side equals the square root of (25 minus 9) = the square root of 16 = 4. Therefore cos(theta) = adjacent divided by hypotenuse = 4/5. This is the classic 3-4-5 right triangle. Option B inverts the fraction. Option C squares the sine value incorrectly. Option D uses the wrong sides.

Option A gives Adjacent divided by Opposite, which is cotangent, not tangent. Option B is correct — sin divided by cos cancels the hypotenuse to give opposite over adjacent. Option C gives 1 divided by sin, which is cosecant, not tangent. Option D is the direct SOHCAHTOA definition of tangent.

sin(theta) = 5/13 means the opposite side is 5 and the hypotenuse is 13. Using the Pythagorean theorem, the adjacent side equals the square root of (169 minus 25) = the square root of 144 = 12. Therefore tan(theta) = opposite divided by adjacent = 5 divided by 12 = 5/12. This is the 5-12-13 right triangle. Option B inverts the ratio. Options C and D involve the hypotenuse instead of the adjacent side.

sin(theta) = 4/5 means the opposite side is 4 and the hypotenuse is 5. Using the Pythagorean theorem, the adjacent side equals the square root of (25 minus 16) = the square root of 9 = 3. Therefore tan(theta) = opposite divided by adjacent = 4 divided by 3 = 4/3. This is the 3-4-5 right triangle. Option A gives the inverse ratio, which is cotangent. Option C uses the hypotenuse instead of the adjacent side. Option D squares the sine value incorrectly.

The answer is True. For an acute angle less than 45 degrees, the adjacent side is longer than the opposite side in the corresponding right triangle. Since cosine uses the adjacent side as its numerator and sine uses the opposite side, and both are divided by the same hypotenuse, cosine produces a larger value than sine. At exactly 45 degrees they become equal. Above 45 degrees the opposite side exceeds the adjacent side and sine becomes larger.

theta = arccos(0.8) = 36.87 degrees. This corresponds to the 3-4-5 right triangle where the adjacent side is 4 and the hypotenuse is 5, giving cos(theta) = 4/5 = 0.8. Option A gives arccos(0.6) = 53.13 degrees, where cosine is 0.6 not 0.8. Option B gives cos(45 degrees) = sqrt(2)/2 approximately 0.707. Option D gives cos(60 degrees) = 0.5. Only 36.87 degrees satisfies the equation.

Option A gives the cosecant of theta, the reciprocal of sine, not cosine. Option B is correct — cosine and sine of complementary angles are equal, so cos(theta) = sin(90 degrees minus theta). Option C inverts the cosine ratio, giving secant. Option D is the direct CAH definition, confirming cosine equals adjacent divided by hypotenuse.