Right-triangle trigonometry — Digital SAT Math practice
Sine, cosine, tangent, complementary angles, radians.
Key terms
- Opposite side: The leg of a right triangle directly across from the given angle.
- Adjacent side: The leg of a right triangle that forms the given angle with the hypotenuse.
- Hypotenuse: The longest side of a right triangle, always opposite the right angle.
- Complementary angles: Two angles whose measures add up to exactly ninety degrees.
- Radian: A unit of angular measure where one full circle equals two pi radians.
Common traps
- Mixing up opposite and adjacent sides when the reference angle changes.
- Using the Pythagorean identity with two different angles; it holds for the same angle only.
- Treating tan as hypotenuse over adjacent; that is sec. Tangent is opposite over adjacent.
Key ideas to know
Calculating Trig Ratios in Right Triangles
In right triangle ABC with angle C = 90 degrees, side a = 3 (opposite A), side b = 4 (adjacent A), and hypotenuse c = 5. What is the value of cos(A)?
4/5. Cosine equals adjacent over hypotenuse relative to angle A. Here, side b is adjacent to A and c is the hypotenuse, so cos(A) = 4/5.
Watch out: Mixing up adjacent and opposite sides or using the wrong ratio for sine, cosine, or tangent.
Complementary Angle Trig Identity
If sin(35 degrees) = k, what is the value of cos(55 degrees)?
k. For complementary angles that sum to 90 degrees, sin(x) equals cos(90 - x). Since 35 + 55 = 90, cos(55) = sin(35) = k.
Watch out: Assuming sin(x) always equals cos(x). They only match when x and its complement have equal ratios, which happens at 45 degrees, not generally.
Converting Degrees to Radians
Convert 75 degrees into radians. Show your conversion factor and final simplified answer.
Multiply by pi/180. 75 * (pi/180) = (75pi)/180 = 5pi/12 radians. Always multiply degree measures by pi/180 to get radians.
Watch out: Dividing by pi/180 instead of multiplying, or forgetting to simplify the fraction after conversion.
Similar triangles share ratios
Right triangles T1 and T2 are similar. In T1, sin(A) = 3/5. If T2 has hypotenuse 20, how long is the side opposite the matching angle A?
12. Similar triangles have the same trigonometric ratios, so opposite/20 = 3/5, which gives 12.
Watch out: Thinking ratios change when the triangle is scaled. They depend only on the angle, not the size.
Identifying Opposite and Adjacent Sides
A student writes tan(40) = opposite / side_next_to_90_degrees. What is wrong with this step and how do you fix it?
Incorrect. The adjacent side must touch the target angle, not the right angle. Fix: Divide the length of the side opposite 40 by the leg touching 40.
Watch out: Using the side next to the right angle as adjacent instead of the side touching your target angle. Adjacent and opposite switch roles when you change reference angles.
Sample questions
Question 1
If sin 36° = cos y° and 0 < y < 90, what is the value of y?
- 144
- 126
- 54
- 36
Show answer and explanation
Answer: 54
sin θ = cos(90° − θ), so y = 90 − 36 = 54.
Question 2
In a right triangle, an acute angle theta has its opposite side equal to 3 and hypotenuse equal to 5. What is cos(theta)?
- 3/5
- 4/5
- 3/4
- 5/4
Show answer and explanation
Answer: 4/5
The adjacent side is sqrt(5^2 - 3^2) = 4, so cos(theta) = adjacent/hypotenuse = 4/5.