4.3 Right-Triangle Trigonometry and Identities

What ACT Trig Usually Tests

ACT lists trigonometric ratios inside the Geometry subcategory, so trig is not a separate marathon topic on this exam. It appears as a tool for missing sides, missing angles, unit-circle values, and occasional identity manipulation. The key is not to memorize every identity from precalculus. The key is to recognize the triangle or circle structure quickly and avoid mixing up which side is opposite or adjacent.

Begin with the angle named in the problem. In a right triangle, the hypotenuse is always opposite the right angle and is the longest side. The opposite side is across from the chosen acute angle. The adjacent side touches the chosen acute angle but is not the hypotenuse. If you switch to the other acute angle, opposite and adjacent swap roles. Many wrong answers come from using the correct ratio for the wrong angle.

SOH-CAH-TOA is the compact map. Sine = opposite/hypotenuse. Cosine = adjacent/hypotenuse. Tangent = opposite/adjacent. If the problem asks for a side, write the ratio equation before solving. If the problem asks for an angle, use inverse trig only after setting the correct ratio. For example, if a ladder is the hypotenuse, the wall height is opposite the ground angle, and the ground distance is adjacent, tangent relates height to distance while sine relates height to ladder length.

Right-Triangle Decision Table

Special triangles are ACT time savers. A 45-45-90 triangle has equal legs, and the hypotenuse is leg times sqrt(2). A 30-60-90 triangle has sides x, x sqrt(3), and 2x, where x is opposite 30 degrees and 2x is the hypotenuse. These ratios produce common trig values: sin 30 degrees = 1/2, cos 60 degrees = 1/2, tan 45 degrees = 1, sin 45 degrees = sqrt(2)/2, and cos 30 degrees = sqrt(3)/2. You can derive them from the triangle instead of memorizing isolated values.

Unit Circle, Radians, and Signs

The unit circle is a circle of radius 1 centered at the origin. A point on the unit circle at angle theta has coordinates (cos theta, sin theta). That means cosine is the x-coordinate and sine is the y-coordinate. This one fact organizes the signs: in Quadrant I both are positive, in Quadrant II sine is positive and cosine is negative, in Quadrant III both are negative, and in Quadrant IV cosine is positive while sine is negative. Tangent, which is sine divided by cosine, is positive in Quadrants I and III.

Radians are another common source of lost points. The conversion is based on pi radians = 180 degrees. To convert degrees to radians, multiply by pi/180. To convert radians to degrees, multiply by 180/pi. Some values should be automatic: 30 degrees = pi/6, 45 degrees = pi/4, 60 degrees = pi/3, 90 degrees = pi/2, 180 degrees = pi, and 360 degrees = 2pi. If a calculator is used, check whether it is in degree or radian mode before using sine, cosine, or tangent.

Identity Toolkit

The most important identity is sin^2(theta) + cos^2(theta) = 1. It is the Pythagorean theorem on the unit circle because x^2 + y^2 = 1. If cos theta = 3/5 and theta is acute, then sin^2 theta = 1 - 9/25 = 16/25, so sin theta = 4/5. The acute-angle condition matters because sine is positive there; in another quadrant, the sign could change.

Another useful identity is tan theta = sin theta / cos theta. If sine is 4/5 and cosine is 3/5, tangent is (4/5)/(3/5) = 4/3. You can also draw a right triangle with adjacent 3, opposite 4, and hypotenuse 5. Drawing the triangle is often less error-prone than manipulating fractions in your head.

Reciprocal trig functions such as cosecant, secant, and cotangent are less central for ACT Math, but they can appear in advanced review or answer choices. Cosecant is 1/sine, secant is 1/cosine, and cotangent is 1/tangent. If these show up, translate them back into sine, cosine, and tangent before solving.

Problem-Solving Patterns and Traps

A reliable ACT trig routine has four steps. First, identify the reference angle. Second, label opposite, adjacent, and hypotenuse from that angle. Third, choose the ratio or identity that contains the known and unknown quantities. Fourth, solve and check the size of the answer. A sine or cosine value must be between -1 and 1, a side length must be positive, and the hypotenuse must be longer than either leg.

For word problems, draw the triangle even if the problem does not provide one. Angles of elevation and depression are measured from a horizontal line. If a person looks up at a tower, the angle of elevation is at the person; if a plane looks down toward a runway, parallel horizontal lines make the matching angle at the ground. Once that angle is placed, tangent usually connects height and horizontal distance, while sine or cosine uses the line of sight.

For identity problems, rebuild the triangle when possible. If sin theta = 5/13 and theta is acute, a 5-12-13 triangle gives cos theta = 12/13. If the question asks for cos squared theta, then 144/169 is appropriate. ACT distractors may include both, so read the requested expression exactly.

Common Traps

• Labeling opposite and adjacent from the wrong acute angle.
• Treating the side across from the right angle as adjacent instead of hypotenuse.
• Leaving a calculator in radian mode for a degree question, or degree mode for a radian question.
• Dropping negative signs for sine or cosine outside Quadrant I.
• Reporting sin squared theta when the problem asks for sin theta.
• Forgetting tangent is undefined when cosine is 0.

The best final check is reasonableness. If an acute angle has a tiny opposite side and a large adjacent side, tangent should be small; if the hypotenuse is 10 and you calculate a leg of 12, the setup used the wrong side or ratio.