Trigonometry Tutoring

Confusing the hypotenuse with another side

The hypotenuse is always opposite the right angle and is the longest side of a right triangle. In a rotated diagram, students may point to a leg instead. Tutors can ask students to locate the 90-degree angle first, then identify the side across from it.

Labeling sides without reference to the angle

Opposite and adjacent are defined relative to a specific acute angle. If the angle changes, the labels change. Tutors can have students mark the angle they are using, then label the side across from it as opposite and the side next to it—other than the hypotenuse—as adjacent.

Choosing the wrong trigonometric ratio

Students may set up sine when the problem involves adjacent and hypotenuse, which requires cosine. Tutors can prompt students to write down which two sides are known or needed, then match those sides to the correct ratio using SOH CAH TOA.

Mixing up the numerator and denominator

In sine equals opposite over hypotenuse, the opposite side goes in the numerator. Reversing the fraction leads to an incorrect ratio. Tutors can ask students to state the ratio aloud and check the setup against the mnemonic before calculating.

Forgetting to use inverse functions when solving for an angle

If sine of an angle equals a known value, students need inverse sine to find the angle measure. Tutors can show that applying the inverse function to both sides isolates the variable, similar to using square roots or logarithms in other equations.

Locate the right angle and hypotenuse before labeling other sides

The first step in any right-triangle trigonometry problem is finding the 90-degree angle. The hypotenuse is the side opposite that angle. Once the hypotenuse is identified, students can label the remaining sides as opposite or adjacent relative to the angle they are working with. This order prevents confusion when the triangle is rotated or drawn in an unfamiliar orientation.

1. Find the right angle in the triangle.
2. Identify the hypotenuse as the side opposite the right angle.
3. Mark the acute angle you will use.
4. Label the side across from that angle as opposite and the remaining side as adjacent.

Match the known and unknown sides to the correct ratio

Students can list which sides are given and which side or angle they need to find. If the problem involves opposite and hypotenuse, use sine. If it involves adjacent and hypotenuse, use cosine. If it involves opposite and adjacent, use tangent. Writing this match explicitly helps students choose the ratio before setting up the equation.

Use inverse trigonometric functions to solve for an angle

When the unknown is an acute angle in a right triangle, an inverse trigonometric function can recover it from a side ratio. If sin A = 0.5, then A = arcsin(0.5) = 30 degrees. Tutors can check calculator mode and explain that the acute-angle restriction makes this answer appropriate; a general sine equation can have other solutions.

From a labeled triangle to the correct ratio

Draw a right triangle with a 90-degree angle at C, an acute angle at A, and sides labeled: hypotenuse 10, opposite 6, adjacent 8. To find angle A, use sine equals opposite over hypotenuse, so sine A equals 6 over 10 or 0.6. Then A equals inverse sine of 0.6, approximately 37 degrees. Students can explain why sine was chosen and check that the opposite side was used in the numerator.

From a word problem to a trigonometric equation

A ladder 13 feet long leans against a wall, reaching 12 feet high. Find the angle the ladder makes with the ground. Draw the triangle: the ladder is the hypotenuse, the wall is opposite the ground angle, and the ground is adjacent. Use sine equals opposite over hypotenuse: sine of the angle equals 12 over 13. The angle equals inverse sine of 12 over 13, approximately 67 degrees. Tutors can ask students to identify which side is which before choosing the ratio.