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Trig Functions – Learning Outcomes

Learn to solve trig function problems in triangles, using Pythagoras' theorem, in all quadrants, and for angles over 360°. Understand the ratios of sides and inverse trig functions. Practice calculating and drawing triangles for real-world applications.

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Trig Functions – Learning Outcomes

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  1. Trig Functions – Learning Outcomes • Solve problems about trig functions in right-angled triangles. • Solve problems using Pythagoras’ theorem. • Solve problems about trig functions in all quadrants of a unit circle. • Solve problems about trig functions of angles greater than 360o and less than 0o.

  2. pg 260-263 Use Trig Functions (Right-Angled Triangles) • Recall in a right-angled triangle, we name three sides based on the angle of interest: • “Hypotenuse” is always the longest side. • “Opposite” is literally opposite the angle of interest. • “Adjacent” is literally adjacent to the angle of interest.

  3. pg 260-263 Use Trig Functions (RAT) • For a particular angle, the ratio between the side lengths is the same for every triangle, regardless of size. • The ratio of the opposite to the hypotenuse is called : • The ratio of the adjacent to the hypotenuse is called : • The ratio of the opposite to the adjacent is called :

  4. pg 260-263 Use Trig Functions (RAT) • These are given on page 13 of the Formula and Tables Book, but are presented without the words “opposite”, “adjacent”, or “hypotenuse”.

  5. pg 260-263 Use Trig Functions (RAT) • e.g. Calculate , , and using the triangle below:

  6. pg 260-263 Use Trig Functions (RAT) • e.g. Use a calculator to calculate each of the following and fill in the blanks. • , so the opposite is 0.866 times as long as the hypotenuse. • , so the is times as long as the . • , so the is times as long as the . • , so the is times as long as the .

  7. pg 260-263 Use Trig Functions (RAT) • Draw a triangle so that , , and . Find and . • Draw a triangle so that , , and . Find and . • Draw a triangle so that , , and . Find and .

  8. pg 260-263 Use Trig Functions (RAT) • While the trig functions apply to angles and return the ratio of side lengths, there are inverse trig functions that apply to ratios and return angles. • If , then • If , then • If , then • On a calculator, these are typed using e.g. SHIFT+sin • is pronounced “inverse sine of B” or “sine minus one of B”, likewise for and .

  9. pg 260-263 Use Trig Functions (RAT) • e.g. Given the triangle shown, find :

  10. pg 260-263 Use Trig Functions (RAT) • Draw a triangle so that , , and . Find . • Draw a triangle so that , , and . Find . • Draw a triangle so that , , and . Find .

  11. pg 260-263 Use Trig Functions (RAT) • In real world problems, two addition terms are important. • When looking up, the angle from the horizontal is called the angle of elevation. • When looking down, the angle from the horizontal is called the angle of depression. • Additionally, a clinometer is a device used to measure these angles.

  12. pg 260-263 Use Trig Functions (RAT) • From the top of a light house 60 meters high with its base at the sea level, the angle of depression of a boat is 15 degrees. What is the distance of boat from the foot of the light house? • The angle of elevation of the top of an incomplete vertical pillar at a horizontal distance of 100 m from its base is 45 degrees. If the angle of elevation of the top of the complete pillar at the same point is to be 60 degrees, then the height of the incomplete pillar is to be increased by how much ? • A 10 meter long ladder rests against a vertical wall so that the distance between the foot of the ladder and the wall is 2 meter. Find the angle the ladder makes with the wall and height above the ground at which the upper end of the ladder touches the wall.

  13. pg 262 Use Pythagoras’ Theorem • Pythagoras’ Theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

  14. pg 262 Use Pythagoras’ Theorem • Find the missing side length in each of the following triangles. 1. 2. 3.

  15. pg 264-268 Solve Quadrant Problems • The right-angled triangle definitions of sine, cosine, and tangent limit the possible angles between 0 and 90 degrees. • To generalise the trig functions, we redefine them based on the unit circle. • The unit circle is a circle on the coordinate plane with centre (0, 0) and radius 1.

  16. pg 264-268 Solve Quadrant Problems • Measuring the angle anti-clockwise from the positive x-axis: • is the x-coordinate where the radius meets the circumference, • is the y-coordinate where the radius meets the circumference.

  17. pg 264-268 Solve Quadrant Problems • Using the unit circle, complete this table. • Recall that angles measure anti-clockwise from the positive x-axis. • is the x-coordinate • is the y-coordinate

  18. pg 264-268 Solve Quadrant Problems • Draw a unit circle including axes. • Draw radiuses at 60o, 120o, 240o, and 300o. • Find for each of the angles drawn. • Find for each of the angles drawn.

  19. pg 264-268 Solve Quadrant Problems • Each angle can be reframed as the smallest angle made with the x-axis, called the reference angle. • For 60o, 120o, 240o, and 300o, this angle is 60o. • Since , cosine of each of the other angles is either 0.5 or -0.5 depending on what side of the circle it’s on.

  20. pg 264-268 Solve Quadrant Problems • Recall that represents the x-coordinate of the circumference. • Thus, it is positive in the first and fourth quadrants of the circle. • Similarly, it is negative in the second and third quadrants of the circle.

  21. pg 264-268 Solve Quadrant Problems • Similarly, , so the sine of each of the other angles is either 0.866 or -0.866.

  22. pg 264-268 Solve Quadrant Problems • Recall that represents the y-coordinate of the circumference. • Thus, it is positive in the first and second quadrants of the circle. • Similarly, it is negative in the third and fourth quadrants of the circle.

  23. pg 264-268 Solve Quadrant Problems • Recall that represents . • Thus, it is positive in the first and third quadrants of the circle • Similarly, it is negative in the second and fourth quadrants of the circle. • (You can think of it as the slope of the radius).

  24. pg 264-268 Solve Quadrant Problems • If , find two values of if . • If , find two values of if . • If , find two values of if .

  25. pg 268-269 Solve > 360o and < 0o Problems • Sine, cosine, and tangent are periodic. • Each function repeats itself every 360o. • e.g. . • In general,

  26. pg 268-269 Solve > 360o and < 0o Problems • Likewise, if we measure negative angles (going clockwise instead of anti-clockwise), the result is equivalent to some anti-clockwise angle. • In general, • with negative

  27. pg 268-269 Solve > 360o and < 0o Problems • Find:

  28. pg 268-269 Solve > 360o and < 0o Problems • e.g. If , find all values of if . • , so our reference angle is . • Cosine is positive in the first and fourth quadrants, so:

  29. pg 268-269 Solve > 360o and < 0o Problems • If , find two values of if . • If , find two values of if . • If , find two values of if .

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