Grade 10 Academic (MPM2D) Unit 6: Trigonometry 2: Non-Right Triangles SSA - Triangles Investigations Mr. Choi © 2017 E. Choi – MPM2D - All Rights Reserved

Trigonometry is a branch of mathematics that studies the relationship between the measures of the angles and the lengths of the sides in triangles. SSA Triangles Investigations © 2017 E. Choi – MPM2D - All Rights Reserved

Notation: In any triangle, the vertices are named using capital letters as in the given example. The lengths of the sides are named using lower case letters that match the opposite vertex. SSA Triangles Investigations © 2017 E. Choi – MPM2D - All Rights Reserved

Sine (Sin), COSINE (Cos), TANGENT (Tan): Opposite side Adjacent side Hypotenuse side SSA Triangles Investigations © 2017 E. Choi – MPM2D - All Rights Reserved

Sine Law Cosine Law a Using Cosine Law to find the angles or © 2017 E. Choi – MPM2D - All Rights Reserved SSA Triangles Investigations

Steps to solve triangle Right triangles can be solved with the primary trig. Ratios Non-rights triangles can be solved with sine law & cosine law. Use Sine Law if: - 2 sides and one opposite angle to one of the given sides are known (SSA) - 2 angles and one opposite side to one of the given angles are known (AAS) Use Cosine Law if: - all 3 sides are known (SSS) - 2 sides and a contained angle are known (SAS) Applications of Sine & Cosine Laws © 2017 E. Choi – MPM2D - All Rights Reserved

SINE Law (Alternate possibilities SINE Law (Alternate possibilities!!) Sine Law can be used to compute the remaining sides of a triangle when two angles and a side are known. It can also be used when two sides and one of the non-enclosed angles are known. In some such cases, the triangle is not uniquely determined by this data (called the ambiguous case) and the technique gives two possible values for the enclosed angle. You are going to learn in more detail in the next course. SSA Triangles Investigations © 2017 E. Choi – MPM2D - All Rights Reserved

SSA: when the given angle B is acute Given where point A is on the horizontal, line and a line segment AC. 3 cm is the required length to form a right triangle. SSA Triangles Investigations © 2017 E. Choi – MPM2D - All Rights Reserved

SSA: when the given angle is acute Given Side 3rd Side Given Angle Recommendation: In SSA situation, built the triangle from the horizontal. Start with the Given Angle from the horizontal line. Attach the Given Angle with the Given side, Attach the Given side, with the 3rd side accordingly as shown in the picture above. SSA Triangles Investigations © 2017 E. Choi – MPM2D - All Rights Reserved

SSA: when the given angle B is acute Given where point A is on the horizontal, line and a line segment AC. Given side (6cm) Given angle: 30o Case 1: If AC = 3 cm Start the angle from horizontal line!! 3 cm A Therefore: 1 right triangle can be formed. Given side (6cm) x sin(given angle) = 3rd Side (3 cm) SSA Triangles Investigations © 2017 E. Choi – MPM2D - All Rights Reserved

SSA: when the given angle B is acute Given where point A is on the horizontal, line and a line segment AC. Given side (6cm) Given angle: 30o Case 2: If AC < 3 cm Start the angle from horizontal line!! 2 cm A ie: Let AC = 2 cm Given side (6cm) x sin(given angle) Therefore: No triangle can be formed. > 3rd Side (2 cm) SSA Triangles Investigations © 2017 E. Choi – MPM2D - All Rights Reserved

SSA: when the given angle B is acute Given where point A is on the horizontal, line and a line segment AC. Given side (6cm) Given angle: 30o Case 3: If 3cm < AC < 6 cm Start the angle from horizontal line!! 4 cm A’ 4 cm A ie: Let AC = 4 cm Given side (6cm) x sin(given angle) Therefore: Two triangles can be formed. < 3rd Side (4 cm) < Given Side (6cm) SSA Triangles Investigations © 2017 E. Choi – MPM2D - All Rights Reserved

SSA: when the given angle B is acute Given where point A is on the horizontal, line and a line segment AC. Given side (6cm) Given angle: 30o Case 4: If AC = 6 cm Start the angle from horizontal line!! 6 cm A ie: Let AC = 6 cm Therefore: 1 Isosceles triangle can be formed. Given side (6cm) = 3rd Side (6cm) SSA Triangles Investigations © 2017 E. Choi – MPM2D - All Rights Reserved

SSA: when the given angle B is acute Given where point A is on the horizontal, line and a line segment AC. Given side (6cm) Given angle: 30o Case 5: If AC > 6 cm Start the angle from horizontal line!! 8 cm A ie: Let AC = 8 cm Therefore: 1 obtuse triangle can be formed. Given side (6cm) < 3rd Side (8cm) SSA Triangles Investigations © 2017 E. Choi – MPM2D - All Rights Reserved

Example 1: In , d = 64cm, f = 42 cm, and. Solve the triangle Example 1: In , d = 64cm, f = 42 cm, and . Solve the triangle. Round to nearest degrees and nearest tenth - centimeters. Triangle 1 Supplementary Angles 84° 42 cm D 61° 72.8 cm Given side (64cm) x sin(given angle) Two triangles can be formed. < 3rd Side (42 cm) < Given Side (64cm) SSA Triangles Investigations © 2017 E. Choi – MPM2D - All Rights Reserved

Example 1: Continued In , d = 64cm, f = 42 cm, and. Solve the triangle Example 1: Continued In , d = 64cm, f = 42 cm, and . Solve the triangle. Round to nearest degrees and nearest tenth - centimeters. Triangle 2 42 cm D’ 26° 119° 32.1cm Supplementary Angles Given side (64cm) x sin(given angle) Two triangles can be formed. < 3rd Side (42 cm) < Given Side (64cm) SSA Triangles Investigations © 2017 E. Choi – MPM2D - All Rights Reserved

SSA: when the given angle is acute (Summary) Given Side 3rd Side Given Angle Given Side x sin(Given Angle) = 3rd Side  1 Right Triangle Given Side x sin(Given Angle) > 3rd Side  No Triangle Given Side x sin(Given Angle) < 3rd Side < Given Side  2 Triangles Given Side = 3rd Side  1 Isosceles Triangle Given Side < 3rd Side  1 Obtuse Triangle SSA Triangles Investigations © 2017 E. Choi – MPM2D - All Rights Reserved

What if the given angle B is an obtuse angle?? 3rd Side Given Side Given Angle (obtuse) A triangle can be formed only when…. 3rd Side > Given Side Will be discussed in next course! SSA Triangles Investigations © 2017 E. Choi – MPM2D - All Rights Reserved

Homework Work sheet: SSA – Investigations Exercise: #1-4 Text: Check the website for updates SSA Triangles Investigations © 2017 E. Choi – MPM2D - All Rights Reserved

End of lesson Applications of Sine & Cosine Laws © 2017 E. Choi – MPM2D - All Rights Reserved