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Lesson · Grade 11 · Pre Calculus

Angles in Standard Position and Reference Angles

  • Grade 11
  • Pre Calculus
  • 2 min 3 s
  • British Columbia
  • BC11-MA-82

Starting on the positive x-axis and turning counterclockwise to the point (−3, 4), the video finds r = 5, sin θ = 4/5, cos θ = −3/5, tan θ = −4/3, a reference angle of about 53.1°, θ ≈ 126.9° and the coterminal angle 486.9°. The same circle then becomes the unit circle with the special angles, where a 30-60-90 triangle and the sign of sine in quadrants I and II solve sin θ = 1/2 as θ = 30° or 150°, warning against giving only one answer.

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In one sentence

The question this lesson answers

Which angle reaches the point negative 3, 4?

In standard position, the vertex is at the origin and the initial arm is the positive x-axis.

Transcript

What the lesson says

Which angle reaches the point negative 3, 4?

In standard position, the vertex is at the origin and the initial arm is the positive x-axis.

Turn counterclockwise to the terminal arm, through our point in quadrant two.

By Pythagoras, the distance r is the square root of 9 plus 16, which is 5.

Sine is y over r, 4 fifths. Cosine is x over r, negative 3 fifths. Tangent is y over x, negative 4 thirds.

The reference angle is the acute angle to the x-axis: tan inverse of 4 thirds, about 53.1 degrees.

So theta is 180 minus 53.1, about 126.9 degrees. Check: its sine is about 0.8, which is 4 fifths.

Add a full turn of 360 degrees: 486.9 degrees ends on the same arm, so it is coterminal.

Now solve sine theta equals one half, from 0 up to 360 degrees.

On the unit circle, with its special angles, r is 1, so sine theta is just y.

A 30-60-90 triangle has sides 1, root 3 and 2, so sine 30 degrees is one half.

So the reference angle is 30 degrees, and sine is positive in quadrants one and two.

In quadrant one, theta is 30 degrees. In quadrant two, it is 180 minus 30, or 150.

A common mistake is stopping at 30 degrees. Both points have y equal to one half, so both are answers.

Recap: find the reference angle, then use it in every quadrant where the sign fits.

Curriculum

Where this fits

British Columbia, Grade 11

Pre Calculus 11, Trigonometry: trigonometry: non-right triangles and angles in standard positionBC11-MA-82

Previous: The Ambiguous Case: Two Triangles from the Same FactsNext: Exact Values from the Special Triangles

British Columbia, Grade 12

Pre Calculus 12, Trigonometry: trigonometry: functions, equations, and identitiesBC12-MA-180

Next: The Unit Circle and Trig Ratios

This lesson is also listed in Grade 12.

The outcomes and strands follow the British Columbia curriculum.

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